Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields

Research ArticleHermitian Self-Orthogonal Constacyclic Codes over Finite Fields

Amita Sahni and Poonam Trama Sehgal

Centre for Advanced Study in Mathematics Panjab University Chandigarh 160014 India

Correspondence should be addressed to Amita Sahni sahniamita05gmailcom

Received 21 July 2014 Accepted 23 October 2014 Published 12 November 2014

Academic Editor Tamir Tassa

Copyright copy 2014 A Sahni and P T Sehgal This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length 119899 over a finite fieldF1199022 119899 coprime to 119902 are found The defining sets and corresponding generator polynomials of these codes are also characterised A

formula for the number of Hermitian self-orthogonal constacyclic codes of length 119899 over a finite field F1199022is obtained Conditions

for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained The defining set and the numberof such MDS codes are also found

1 Introduction

Let F1199022 denote a finite field with 1199022 elements An [119899 119896]linear code 119862 of length 119899 and dimension 119896 over F1199022 is a 119896-dimensional subspace of the vector space F119899

1199022 Elements of

the subspace 119862 are called codewords and are written as rowvectors 119888 = (1198880 1198881 119888119899minus1) A linear code 119862 over F1199022 iscalled 120582-constacyclic if (120582119888119899minus1 1198880 119888119899minus2) is in 119862 for every(1198880 1198881 119888119899minus1) in 119862 Let 120579 F

119899

1199022rarr F1199022[119909]⟨119909

119899minus 120582⟩ be the

map given by 120579((1198880 1198881 119888119899minus1)) 997891rarr 1198880 + 1198881119909 + sdot sdot sdot + 119888119899minus1119909119899minus1

mod(119909119899 minus 120582) One can easily check that 120579 is an F1199022-moduleisomorphismWe can therefore identify 120582-constacyclic codesof length 119899 over F1199022 with ideals in F1199022[119909]⟨119909

119899minus 120582⟩ The

Hamming weight 119908(119888) of 119888 isin F1198991199022is the number of nonzero

coordinates of 119888 The minimum distance of 119862 is defined to be119889 = min119908(119888) 119888( = 0) isin 119862 An [119899 119896 119889] code that is a [119899 119896]linear code withminimum distance 119889 is said to bemaximumdistance separable (MDS) if 119889 = 119899 minus 119896 + 1 The Hermitianinner product of elements 119906 V isin F119899

1199022is defined as ⟨119906 V⟩119867 =

sum119899minus1

119894=0119906119894V119902

119894 for 119906 = (1199060 1199061 119906119899minus1) and V = (V0 V1 V119899minus1)

For a linear code 119862 of length 119899 over F1199022 the Hermitian dualcode119862perp119867 of119862 is defined by119862perp119867 = V isin F1199022 ⟨119906 V⟩119867 = 0 forall119906 isin119862 If 119862 = 119862perp119867 then 119862 is known as Hermitian self-dual and119862 is Hermitian self-orthogonal if 119862 sube 119862perp119867

Aydin et al [1] dealt with constacyclic codes and aconstacyclic BCH boundwas given Gulliver et al [2] showedthat there exists Euclidean self-dual MDS code of length 119902over F119902 when 119902 = 2

119898 by using a Reed-Solomon (RS) code andits extensionThey also constructedmany new Euclidean andHermitian self-dual MDS codes over finite fields Blackford[3] studied negacyclic codes over finite fields by using multi-pliers He gave conditions on the existence of Euclidean self-dual codes Recently Guenda [4] constructedMDSEuclideanand Hermitian self-dual codes from extended cyclic duadicor negacyclic codes and gave necessary and sufficient con-ditions on the existence of Hermitian self-dual negacycliccodes arising from negacyclic codes In [5] the authors gaveformulae to enumerate the number of Euclidean self-dualand self-orthogonal negacyclic codes of length 119899 over a finitefield F119902 where 119902 is coprime to 119899 In [6] Yang and Caigave the necessary and sufficient conditions for the existenceof Hermitian self-dual constacyclic codes They also gavesome conditions under which Hermitian self-dual and self-orthogonal MDS codes exist In this paper we find necessaryand sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length 119899 over a finite fieldF1199022 119899 coprime to 119902 and also give a characterization of theirdefining sets We obtain a formula to calculate the numberof these codes We give conditions for the existence of someMDS Hermitian self-orthogonal constacyclic codes We alsofound their number and defining sets (Table 1)

Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 985387 7 pageshttpdxdoiorg1011552014985387

2 Journal of Discrete Mathematics

Table 1 Number of Hermitian self-orthogonal codes over F25

119903 119899 119873 |Λ(1)

1|

2

2 2 34 4 96 6 278 4 911 2 312 12 72913 6 2714 6 2716 4 917 2 318 10 24319 2 322 6 2724 12 729

3

4 2 38 6 2711 2 312 2 313 6 2716 10 24317 2 319 2 322 4 924 6 2726 12 72928 6 2731 10 24332 14 2187

6

2 2 34 4 96 2 38 4 911 2 312 4 913 6 2714 6 2716 4 917 2 318 2 319 2 322 6 2724 4 9

2 Hermitian Self-OrthogonalConstacyclic Codes

Let 119902 be an odd prime power and 119899 a positive integer relativelyprime to 119902 Let 119862 be an [119899 119896] 120582-constacyclic code over F1199022with 119903 = ord1199022(120582) where ord1199022(120582) denotes the order of 120582 inF1199022 Let 119892(119909) be the generator polynomial of 119862 Then 119892(119909)divides (119909119899 minus 120582) Write (119909119899 minus 120582) = 119892(119909)ℎ(119909) The polynomial

ℎ(119909) is called the check polynomial of 119862 For 0 le 119904 lt 119903119899 let119862119904 = 119904 119904119902

2 119904(119902

2)119899119904minus1 be the 1199022-cyclotomic coset modulo

119903119899 containing 119904 where 119899119904 is the least positive integer suchthat 119904(1199022)119899119904 equiv 119904(mod119903119899) Let 120572 be a primitive 119903119899 th rootof unity in some extension field of F1199022 such that 120572119899 = 120582Then the polynomial119872(119904)(119909) = prod119894isin119862

119904

(119909 minus 120572119894) is the minimal

polynomial of 120572119904 over F1199022 and

119909119903119899minus 1 = prod

119904isinΛ

119872(119904) (119909) (1)

where Λ is the set of representatives of all the distinct 1199022-cyclotomic cosets modulo 119903119899 As (119909119899 minus 120582) | (119909119903119899 minus 1) one cancheck that the roots of (119909119899 minus 120582) are precisely 120572119894119903+1 0 le 119894 lt 119899Define

119874119903119899 (1) = 119894119903 + 1 0 le 119894 lt 119899 (mod 119903119899) (2)

Hence we have

119909119899minus 120582 = prod

119904isinΛ1

119872(119904) (119909) (3)

where Λ 1 = Λ cap 119874119903119899(1)Let 119862 = ⟨119892(119909)⟩ be a 120582-constacyclic code with defining set

119879 = 119894119903 + 1 isin 119874119903119899(1) 120572119894119903+1 is a root of 119892(119909) Clearly 119879 is a

union of some 1199022-cyclotomic cosets 119862119904 mod 119903119899 for 119904 isin Λ 1The Hermitian dual 119862perp119867 of the code 119862 is a 120582minus119902-constacycliccode over F1199022 with defining set 119879

perp119867= minus119902[119874119903119899(1) 119879]mod 119903119899

(seeTheorem 32 of [6])Write the generator polynomial119892(119909)of the code 119862 as 119892(119909) = prod

119904isinΛ1

(119872(119904)(119909))120575(119904) where

120575(119904) =

1 if 119904 isin 1198790 if 119904 notin 119879

(4)

Then the generator polynomial of the Hermitian dual 119862perp119867 of119862 is ℎperp119867(119909) = prod

119904isinΛ1

(119872(119904)(119909))Δ(119904) where

Δ (119904) =

1 if 119904 isin 119879perp119867 0 if 119904 notin 119879perp119867

(5)

It can be easily verified that Δ (119904) = 1 minus 120575(minus119902119904) so that ℎperp119867(119909) =

prod119904isinΛ1

(119872(119904)(119909))1minus120575(minus119902119904)

Lemma 1 Let 119862 be a 120582-constacyclic code over F1199022 with1199001199031198891199022(120582) = 119903 If 119862 is a Hermitian self-orthogonal code then119903 | (119902 + 1)

Proof The proof is similar to [6 Proposition 23]

Theorem 2 Nontrivial Hermitian self-orthogonal 120582-consta-cyclic codes of length 119899 over F1199022 exist if and only if 119862119904 = 119862minus119902119904for some 119904 isin 119874119903119899(1)

Proof Let 119862 be a nontrivial Hermitian self-orthogonal 120582-constacyclic code of length 119899 over F1199022 with defining set 119879Then there exists 119904 isin 119874119903119899(1) such that 119904 = 0 and 119904 notin 119879

Journal of Discrete Mathematics 3

Hence 119904 isin 119874119903119899(1) 119879 giving us that minus119902119904 isin 119879perp119867 sube 119879 Thus119862119904 = 119862minus119902119904 (as 119904 notin 119879 and minus119902119904 isin 119879) Conversely let 119904 isin 119874119903119899(1)be such that 119862119904 = 119862minus119902119904 Consider 119879 = 119874119903119899(1) 119862119904 Thecode 119862 is a nontrivial Hermitian self-orthogonal code since119879perp119867= minus119902[119874119903119899(1) 119879] = minus119902119862119904 = 119862minus119902119904 sube 119879

Define 1198790 = 119904 isin 119874119903119899(1) 119862119904 = 119862minus119902119904 The followingtheorem characterizes the defining set of a Hermitian self-orthogonal constacyclic code of length 119899 over F1199022

Theorem 3 Let 119862 be a 120582-constacyclic code of length 119899 over F1199022with the defining set 119879 Then 119862 is Hermitian self-orthogonal ifand only if (i) 1198790 sub 119879 and (ii) for each 119904 notin 1198790 at least one of 119904and minus119902119904 belongs to 119879

Proof Let 119862 be a Hermitian self-orthogonal constacycliccode Let 119904 isin 1198790 Then 119862119904 = 119862minus119902119904 Suppose that 119904 notin 119879Then minus119902119904 isin 119879perp119867 sube 119879 so that 119862minus119902119904 sub 119879 and 119862119904 sube 119879 whichcontradicts the hypothesis that119862119904 = 119862minus119902119904Thus1198790 sube 119879 Nowlet 119904 notin 1198790 Then either 119904 isin 119879 or minus119902119904 isin 119879perp119867 sube 119879 as required

Conversely let the defining set 119879 be such that 1198790 sub 119879and for each 119904 notin 1198790 at least one of 119904 and minus119902119904 is in 119879 Then119879perp119867= minus119902[119874119903119899(1) 119879] = minus119902119904 119904 notin 119879 sube 119879 by condition (ii)

so that the code 119862 having 119879 as a defining set is a Hermitianself-orthogonal code

Corollary 4 A 120582-constacyclic code 119862 of length 119899 over F1199022

generated by 119892(119909) = prod119904isinΛ1

(119872(119904)(119909))120575(119904) is a Hermitian self-

orthogonal if and only if 120575(119904) + 120575(minus119902119904) ge 1 for all 119904 isin Λ 1

Define Λ(0)1= Λ 1 cap 1198790 and Λ

(1)

1= Λ 1 Λ

(0)

1 Observe that

1198790 = ⋃119904isinΛ(0)

1

119862119904

Example 5 Let 119902 = 13 119899 = 15 and 119903 = 7 then 1199022 = 169 Weconsider the 120582-constacyclic code of length 15 over F132 where120582 isin F132 with order 7 Clearly

119874715 (1)

= 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

(6)

1198790 = 15 Let 119879 = 1 15 22 36 43 50 64 99 then 119879perp119867=

minus13[119874715(1) 119879] = 1 22 36 43 50 64 99 sub 119879 Hence thecodewith defining set119879 is a [15 7]Hermitian self-orthogonal120582-constacyclic code

Theorem 6 The number of Hermitian self-orthogonal 120582-con-stacyclic codes of length 119899 over F1199022 is 3|Λ

(1)

1|2

Proof Let 119862 be a Hermitian self-orthogonal 120582-constacy-clic code of length 119899 over F1199022 generated by 119892(119909) =

prod119904isinΛ1

(119872(119904)(119909))120575(119904) Then 120575(119904) + 120575(minus119902119904) ge 1 For 119904 isin Λ

(0)

1

120575(119904) = 120575(minus119902119904) = 1 However for 119904 isin Λ(1)

1 the pairs (120575(119904) 120575(minus119902119904))

have three choices (0 1) (1 0) and (1 1) Hence the numberof Hermitian self-orthogonal 120582-constacyclic codes of length119899 over F1199022 is 3

|Λ(1)

1|2

In order to find the number of Hermitian self-orthogonal120582-constacyclic codes of length 119899 over F1199022 we need to computethe value of |Λ(1)

1| Our aim is to prove the following

Theorem 7 Let 119886 119903 and 119889 be positive integers such that119892119888119889(119886 119903) = 1 Then the number of solutions for the linearcongruence

119886119909 equiv 1 (mod 119903) (7)

in the set 119860 = 119908 0 le 119908 lt 119903119889 and 119892119888119889(119908 119903119889) = 1 is exactly120601(119903119889)120601(119903)

Since gcd(119886 119903) = 1 the linear congruence 119886119909 equiv 1(mod119903) has a unique solution modulo 119903 Let it be 119909 equiv 119887(mod 119903)Then gcd(119887 119903) = 1 We write 119860 = 119894119903 + 119904 0 le 119894 lt 119889 0 le119904 lt 119903 gcd(119894119903 + 119904 119903119889) = 1 The solutions of (7) will beamongst 119860119887 = 119894119903 + 119887 0 le 119894 lt 119889 Clearly the elements of119860119887 are relatively prime to 119903 We need to count the numberof elements of 119860119887 which are coprime to 119889 Also |119860119887| = 119889Therefore the required number 119873 = 119889 minus |119894119903 + 119887 0 le 119894 lt119889 gcd(119894119903 + 119887 119889) gt 1|

Lemma 8 Let 119901 be a prime divisor of 119889 such that 119901 ∤ 119903 Thenumber of multiples of 119901 in 119860119887 is 119889119901

Proof Write 119860119887 = ⋃(119889119901)minus1

119896=0119860(119896)

119887 where 119860(119896)

119887= 119896119901119903 + 119887 (119896119901 +

1)119903 + 119887 ((119896 + 1)119901 minus 1)119903 + 119887 for each 119896 0 le 119896 lt 119889119901 Sinceeach119860(119896)

119887contains119901 elementswhich are pairwise incongruent

mod119901 each 119860(119896)119887

forms a complete residue system mod119901Hence exactly one element in each 119860(119896)

119887is divisible by 119901

Consequently there exist 119889119901 elements in 119860119887 which aredivisible by 119901

Lemma 9 Let 1199011 and 1199012 be two distinct prime divisors of 119889with 119892119888119889(11990111199012 119903) = 1 Then the number of multiples of 1199011 or1199012 in 119860119887 is 1198891199011 + 1198891199012 minus 11988911990111199012

Proof Thenumber ofmultiples of1199011 in119860119887 equals1198891199011 whilethe number of multiples of 1199012 in 119860119887 is 1198891199012 By a similarargument as in Lemma 8 the number of multiples of both1199011 and 1199012 equals 11988911990111199012 Therefore the required number is1198891199011 + 1198891199012 minus 11988911990111199012

Theorem 10 Let 1199011 1199012 119901119898 be all the distinct primedivisors of 119889 which are relatively prime to 119903 The number ofelements in 119860119887 which are not coprime to 119889 is

1198730 = sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 119889119894119904119905119894119899119888119905

119889

119901119894119901119895119901119896

+ sdot sdot sdot + (minus1)119898minus1 119889

11990111199012 119901119898

(8)

Proof The proof follows by induction on 119898 Lemmas 8 and9

In order to prove Theorem 7 it is enough to show that119873 = 119889 minus 1198730 = 120601(119903119889)120601(119903)


Now

119873 = 119889 minus

sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 distinct

119889

119901119894119901119895119901119896


11990111199012 sdot sdot sdot 119901119898

= 119889(1 minus

1

1199011

)(1 minus

1

1199012

) sdot sdot sdot (1 minus

1

119901119898

)

(9)

Let 1199011 1199012 119901119898+119897 be all the distinct prime divisors of 119889Also let 119901119898+1 119901119898+119897 119901119898+119897+1 119901119898+119897+119896 be all the distinctprime divisors of 119903 Then (119889 minus 1198730)120601(119903) = 119889119903prod

119898+119897+119896

119894=1(1 minus

(1119901119894)) = 120601(119903119889) which completes the proof of Theorem 7Pick 119904 isin Λ 1 Let gcd(119903119899 119904) = 119898119904 Define Λ 1119898

119904

= 119904 isin

Λ 1 gcd(119903119899 119904) = 119898119904

Theorem 11 Consider

10038161003816100381610038161003816Λ 1119898

119904

10038161003816100381610038161003816=

120601 (119903119889119904)

119900119903119889119903119889119904

(1199022) 120601 (119903)

119894119891 119892119888119889 (119898119904 119903) = 1

0 119894119891 119892119888119889 (119898119904 119903) = 1

(10)

where 119889119904 = 119899119898119904

Proof As 119904 isin Λ 1 119904 equiv 1(mod 119903) so that gcd(119903 119904) = 1 Since119898119904 is a divisor of 119904 we have that gcd(119898119904 119903) = 1Thus |Λ 1119898

119904

| =

0 whenever119898119904 is not coprime to 119903As gcd(119903119899 119904) = gcd(119899 119904) = 119898119904 we write 119904 = 119898119904119904

1015840Also gcd(119889119904 119904

1015840) = gcd(119899119898119904 119904119898119904) = 1 As 119904 isin Λ 1 119904 equiv

1(mod119903) holds giving us that 1198981199041199041015840equiv 1(mod119903) holds with

gcd(119889119904 1199041015840) = 1 and 0 le 1199041015840 lt 119903119889119904 (∵ 0 le 119904 lt 119903119899) ByTheorem 7

there is 120601(119903119889119904)120601(119903) number of elements 119904 = 1198981199041199041015840 in 119874119903119899(1)

satisfying 0 le 1199041015840 lt 119903119889119904 gcd(119889119904 1199041015840) = 1 and119898119904119904

1015840equiv 1(mod 119903)

However we have to calculate the number of such 119904 in Λ 1Now for 119904 isin Λ 1 gcd(119903119899 119904) = 119898119904

1003816100381610038161003816119862119904

1003816100381610038161003816= ord119903119899119898

119904

(1199022) = ord119903119889

119904

(1199022) (11)

Consequently |Λ 1119898119904

| = 120601(119903119889119904)ord119903119889119904

(1199022)120601(119903) whenever

gcd(119898119904 119903) = 1

Define

120594 (119889 119902) =

0 if 119902119896 equiv minus1 (mod 119889) forsome odd integer 119896

1 otherwise(12)

Observe that for 119904 isin Λ 1 119862119904 = 119862minus119902119904 holds if and only if120594(119903119889119904 119902) = 0 where 119889119904 = 119899119898119904 119898119904 = gcd(119904 119899) as definedearlier Recall that for 119904 isin Λ 1 gcd(119898119904 119903) = 1

Theorem 12 One has10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816= sum

119889|119899

119892119888119889(119899119889119903)=1

120594 (119903119889 119902) 120601 (119903119889)

120601 (119903) 119900119903119889rd (1199022)

(13)

Proof The proof follows from Theorem 11 and the abovedefinition

Example 13 (let 119899 = 12 119903 = 3 and 119902 = 5) Then 119874119903119899(1) =1 4 7 10 13 16 19 22 25 28 31 34 1198621 = 1 25 13 1198624 =4 28 16 1198627 = 7 31 19 and 11986210 = 10 34 22 Take Λ 1 =1 4 7 10 Thus Λ(1)

1= 1 7 (as 1198624 = 119862minus20 and 11986210 = 119862minus50)

so that |Λ(1)1| = 2 By Theorem 12 119889 = 3 6 12 are possible

values of 119889 on right hand side Now 120594(9 5) = 120594(18 5) = 0as 53 equiv minus1(mod 18) However 120594(36 5) = 1 as there does notexist any odd integer 119896 such that 5119896 equiv minus1(mod 36) so that

10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816=

120594 (36 5) 120601 (36)

120601 (3) ord36 (25)=

12

3 times 2

= 2 (14)

We will now investigate the behavior of the function120594(119889 119902)

Lemma 14 Let 1198981 and 1198982 be two integers coprime to 119902 suchthat 1199021198961 equiv minus1(mod1198981) and 1199021198962 equiv minus1(mod1198982) for someodd integers 1198961 and 1198962 If 119902 equiv minus1(mod2119886+119887) where 21198861198981and 21198871198982 then there exists an odd integer 119896 such that 119902119896 equivminus1(mod11989811198982)

Proof Write 11989811198982 = 2119886+119887prod119898

119894=1119901119890119894

119894 119901119894 being odd distinct

primes 119890119894 ge 1 Let 119901 be an odd prime divisor of 11989811198982so that there exists an odd integer 119905(= 1198961 119900119903 1198962) such that119902119905equiv minus1(mod119901) Therefore 1199022119905 equiv 1(mod119901) Consequently

ord119901(119902) | 2119905 but ord119901(119902) ∤ 119905 giving us that 2ord119901(119902) For119890 ge 1 we have 2ord119901119890(119902) since ord119901119890(119902) = 119901

119891ord119901(119902) forsome positive integer 119891 Hence 119902119896119901119890 equiv minus1(mod119901119890) where119896119901119890 = ord119901119890(119902)2 is odd Thus

119902119896equiv minus1 (mod11989811198982) (15)

where 119896 = 119897119888119898119896119901119894119890119894

119898

119894=1is odd

Lemma 15 Let 119886 ge 2 Then 119902119896 equiv minus1(mod 2119886) holds for someinteger 119896 ge 1 if and only if 119902 equiv minus1(mod 2119886) In fact such a 119896 isodd

Proof Proof is trivial

Theorem 16 120594(119903119889 119902) = 0 if and only if 119902 equiv minus1(mod2119886+119887)and 2119900119903119889119901(119902) for all odd prime divisors 119901 of 119889 where 2119886119889and 2119887119903

Proof If 120594(119903119889 119902) = 0 then there exists an odd integer 119896such that 119902119896 equiv minus1(mod 119903119889) Thus 119902119896 equiv minus1(mod 2119886+119887) sothat by Lemma 15 119902 equiv minus1(mod 2119886+119887) Also 119902119896 equiv minus1(mod 119901)for every odd prime divisor 119901 of 119889 Thus 1199022119896 equiv 1(mod119901) showing that ord119901(119902)|2119896 and ord119901(119902) ∤ 119896 Therefore2ord119901(119902) for all odd prime divisors 119901 of 119889

Conversely let 119902 equiv minus1(mod2119886+119887) and 2ord119901(119902) for allodd prime divisors 119901 of 119889 To prove 120594(119903119889 119902) = 0 we need tofind an odd integer 119896 such that 119902119896 equiv minus1(mod119903119889) For anyodd prime divisor 119901 of 119889 as 2ord119901(119902) 119902

119896119901equiv minus1(mod119901)


where 119896119901 = ord119901(119902)2 is odd As in the proof of Lemma 14there exists an odd integer 119896119889 such that 119902119896119889 equiv minus1(mod1198891015840)where 119889 = 21198861198891015840 Also 119902 equiv minus1(mod2119886) Therefore 119902119896119889 equivminus1(mod 119889) with 119896119889 odd By Lemma 1 119902 equiv minus1(mod 119903) UsingLemma 14 we get that 119902119896 equiv minus1(mod 119903119889) for some oddinteger 119896

Proposition 17 If 119903 and 119899 are coprime then120594(119903119889 119902) = 120594(119889 119902)for all divisors 119889 of 119899

Proof Since 119902 equiv minus1(mod 119903) and gcd(119903 119889) = 1 for some oddinteger 119896 119902119896 equiv minus1(mod 119903119889) if and only if 119902119896 equiv minus1(mod119889)

Corollary 18 There does not exist any nontrivial Hermitianself-orthogonal 120582-constacyclic code of length 119899 over F1199022 if andonly if 119902 equiv minus1(mod2119886+119887) and 2119900119903119889119901(119902) for all odd primedivisors 119901 of 119899 where 2119886119899 and 2119887119903 for 119903 = 1199001199031198891199022(120582)

Proof The proof follows easily fromTheorems 12 and 16

For 119899 odd we have 119886 = 0 The condition 119902 equiv minus1(mod2119886+119887) reads as 119902 equiv minus1(mod2119887) which is always true as 119903 |

(119902 + 1) Hence we have the following

Corollary 19 There does not exist any nontrivial Hermitianself-orthogonal 120582-constacyclic code of odd length 119899 over F1199022 ifand only if 2119900119903119889119901(119902) for all prime divisors 119901 of 119899

3 MDS Hermitian Self-OrthogonalConstacyclic Codes Over F

1199022

Let 119862 be a 120582-constacyclic code of length 119899 over F1199022 andord1199022(120582) = 119903 Let 120572 be a primitive 119903119899 th root of unity in someextension field of F1199022 such that 120572119899 = 120582 Then roots of 119862 are ofthe form 120572119894119903+1 0 le 119894 le 119899 minus 1 Put 120577 = 120572119903

Theorem20 Let the generator polynomial of119862 have roots thatinclude the set 120572120577119894 1198941 +1 le 119894 le 1198941 +119889minus1 Then the minimumdistance of 119862 is at least 119889

Proof See [1 Theorem 22]

By Lemma 1 119903 | (119902 + 1) Write 119902 + 1 = 1199031199040

Theorem21 Let 119899 be a divisor of (119902minus1) Let119879 = 119874119903119899(1)119879119897119898where 119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) for each 119898 le lfloor(119899 minus1 minus 1199040)2rfloor and each 119897 ge 1198970 with

1198970 =

minuslfloor

1199040 minus 1

2

rfloor 119894119891 119899 119894119904 119890V119890119899

minus lfloor

1199040 minus 2

2

rfloor 119894119891 119899 119894119904 119900119889119889

(16)

Then the code 119862 with defining set 119879 is a Hermitian self-orthogonal 120582-constacyclic MDS code with parameters [119899119898 minus119897 + 1 119899 minus 119898 + 119897]

Proof Let 119897119898 be as above and

119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) (17)

Each 119879119897119898 has 119898 minus 119897 + 1 elements If 119862 denotes the code withdefining set 119879 = 119874119903119899(1) 119879119897119898 then the dimension of 119862dim119862 = 119898 minus 119897 + 1 Let 119868 = 119894 119897 le 119894 le 119898 Then the set 119868 =0 1 (119899minus1)119868 has 119899minus(119898minus119897+1) = 119899minus119898+119897minus1 consecutiveelementsmodulo 119899 ByTheorem 20 theminimumdistance of119862 is at least (119899 minus 119898 + 119897) However using the singleton boundthe minimum distance is at most (119899 minus 119898 + 119897) Consequentlythe minimum distance of 119862 equals (119899 minus119898+ 119897) proving that 119862is an MDS code

In order to prove that 119862 is self-orthogonal it is enough toprove that 119879119897119898 cap (minus119902119879119897119898) = 0 We have minus119902(119894119903 + 1) equiv (119899 minus119894 minus 1199040)119903 + 1(mod119903119899) Let 1198681 = 119894 119897 le 119894 le 119898(mod119899) and1198682 = 119899 minus 119894 minus 1199040 119894 isin 1198681(mod 119899) Then 119879119897119898 cap (minus119902119879119897119898) = 0 ifand only if 1198681 cap 1198682 = 0 Let 119894 isin 1198681 cap 1198682 then 119894 = 119899 minus 119895 minus 1199040 forsome 119895 isin 1198681 so that

1199040 + 119894 + 119895 = 119899 equiv 0 (mod 119899) (18)

As 119894 119895 isin 1198681 1198970 le 119897 le 119894 119895 le 119898 le lfloor(119899 minus 1199040 minus 1)2rfloor Thusminus1199040 lt 21198970 le 2119897 le 119894 + 119895 le 2lfloor(119899 minus 1199040 minus 1)2rfloor le 119899 minus 1199040 minus 1 lt 119899 minus 1199040

so that

0 lt 1199040 + 119894 + 119895 lt 119899 (19)

Consequently there does not exist any 119894 119895 such that both (18)and (19) hold thereby showing that 119862 is a Hermitian self-orthogonal constacyclic MDS code

Remark 22 For 119899 even 119897 = 1198970 = minuslfloor(1199040 minus 1)2rfloor and119898 = lfloor(119899 minus1199040 minus 1)2rfloor the codes obtained from above theorem are thesame as given byTheorem 43 of [6]

Proposition 23 Let 119873 be the number of Hermitian self-orthogonal 120582-constacyclic MDS codes of length 119899 which can beobtained from above theorem Then119873 is given by

119873 =

119899 (119899 minus 2)

8

119894119891 119904 119894119904 119890V119890119899 119899 119890V119890119899

119899 (119899 + 2)

8

119894119891 119904 119894119904 119900119889119889 119899 119890V119890119899(119899 minus 1) (119899 + 1)

8

119894119891 119904 119894119904 119890V119890119899 119899 119900119889119889

(119899 minus 3) (119899 minus 1)

8

119894119891 119904 119894119904 119900119889119889 119899 119900119889119889

(20)

Proof The required number 119873 equals the number of 119879119897119898 =119894119903+1 119897 le 119894 le 119898(mod 119903119899) that is we need to select119898minus119897+1consecutive integers from the set 119894 1198970 le 119894 le 1198980 = lfloor(119899 minus1 minus 1199040)2rfloor The number of ways of selecting 119905 consecutiveintegers from the set 119894 119890 le 119894 le 119891 is (119891 minus 119890 minus 119905 + 2) Thus thenumber of such codes is119873 = sum1198980minus1198970+1

119895=1119895 = (1198980 minus 1198970 + 1)(1198980 minus

1198970 + 2)2 Hence the result follows

Tables 2 and 3 list [119899 119896 119889] Hermitian self-orthogonalMDS codes over F1199022 for 119902 le 11 Here 119899 119896 and 119889 denoterespectively the length dimension and minimum distanceof the code while 119879 denotes the defining set 119873 denotes thenumber of such Hermitian self-orthogonal MDS codes


Table 2 [119899 119896 119889]Hermitian self-orthogonal MDS codes over F1199022

119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] N Aydin I Siap and D K Ray-Chaudhuri ldquoThe structure of1-generator quasi-twisted codes and new linear codesrdquo DesignsCodes and Cryptography vol 24 no 3 pp 313ndash326 2001

[2] T A Gulliver J-L Kim and Y Lee ldquoNew MDS or near-MDSself-dual codesrdquo IEEE Transactions on Information Theory vol54 no 9 pp 4354ndash4360 2008

[3] T Blackford ldquoNegacyclic duadic codesrdquo Finite Fields and TheirApplications vol 14 no 4 pp 930ndash943 2008

[4] K Guenda ldquoNew MDS self-dual codes over finite fieldsrdquoDesigns Codes and Cryptography vol 62 no 1 pp 31ndash42 2012

[5] A Sahni and P T Sehgal ldquoEnumeration of self-dual and self-orthogonal negacyclic codes over finite fieldsrdquo In press

[6] Y Yang andW Cai ldquoOn self-dual constacyclic codes over finitefieldsrdquo Designs Codes and Cryptography 2013

[7] H Q Dinh and S R Lopez-Permouth ldquoCyclic and negacycliccodes over finite chain ringsrdquo IEEE Transactions on InformationTheory vol 50 no 8 pp 1728ndash1744 2004

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Number of Hermitian self-orthogonal codes over F25

119903 119899 119873 |Λ(1)

1|

2

2 2 34 4 96 6 278 4 911 2 312 12 72913 6 2714 6 2716 4 917 2 318 10 24319 2 322 6 2724 12 729

3

4 2 38 6 2711 2 312 2 313 6 2716 10 24317 2 319 2 322 4 924 6 2726 12 72928 6 2731 10 24332 14 2187

6

2 2 34 4 96 2 38 4 911 2 312 4 913 6 2714 6 2716 4 917 2 318 2 319 2 322 6 2724 4 9

2 Hermitian Self-OrthogonalConstacyclic Codes

Let 119902 be an odd prime power and 119899 a positive integer relativelyprime to 119902 Let 119862 be an [119899 119896] 120582-constacyclic code over F1199022with 119903 = ord1199022(120582) where ord1199022(120582) denotes the order of 120582 inF1199022 Let 119892(119909) be the generator polynomial of 119862 Then 119892(119909)divides (119909119899 minus 120582) Write (119909119899 minus 120582) = 119892(119909)ℎ(119909) The polynomial

ℎ(119909) is called the check polynomial of 119862 For 0 le 119904 lt 119903119899 let119862119904 = 119904 119904119902

2 119904(119902

2)119899119904minus1 be the 1199022-cyclotomic coset modulo

119903119899 containing 119904 where 119899119904 is the least positive integer suchthat 119904(1199022)119899119904 equiv 119904(mod119903119899) Let 120572 be a primitive 119903119899 th rootof unity in some extension field of F1199022 such that 120572119899 = 120582Then the polynomial119872(119904)(119909) = prod119894isin119862

119904

(119909 minus 120572119894) is the minimal

polynomial of 120572119904 over F1199022 and

119909119903119899minus 1 = prod

119904isinΛ

119872(119904) (119909) (1)

where Λ is the set of representatives of all the distinct 1199022-cyclotomic cosets modulo 119903119899 As (119909119899 minus 120582) | (119909119903119899 minus 1) one cancheck that the roots of (119909119899 minus 120582) are precisely 120572119894119903+1 0 le 119894 lt 119899Define

119874119903119899 (1) = 119894119903 + 1 0 le 119894 lt 119899 (mod 119903119899) (2)

Hence we have

119909119899minus 120582 = prod

119904isinΛ1

119872(119904) (119909) (3)

where Λ 1 = Λ cap 119874119903119899(1)Let 119862 = ⟨119892(119909)⟩ be a 120582-constacyclic code with defining set

119879 = 119894119903 + 1 isin 119874119903119899(1) 120572119894119903+1 is a root of 119892(119909) Clearly 119879 is a

union of some 1199022-cyclotomic cosets 119862119904 mod 119903119899 for 119904 isin Λ 1The Hermitian dual 119862perp119867 of the code 119862 is a 120582minus119902-constacycliccode over F1199022 with defining set 119879

perp119867= minus119902[119874119903119899(1) 119879]mod 119903119899

(seeTheorem 32 of [6])Write the generator polynomial119892(119909)of the code 119862 as 119892(119909) = prod

119904isinΛ1

(119872(119904)(119909))120575(119904) where

120575(119904) =

1 if 119904 isin 1198790 if 119904 notin 119879

(4)

Then the generator polynomial of the Hermitian dual 119862perp119867 of119862 is ℎperp119867(119909) = prod

119904isinΛ1

(119872(119904)(119909))Δ(119904) where

Δ (119904) =

1 if 119904 isin 119879perp119867 0 if 119904 notin 119879perp119867

(5)

It can be easily verified that Δ (119904) = 1 minus 120575(minus119902119904) so that ℎperp119867(119909) =

prod119904isinΛ1

(119872(119904)(119909))1minus120575(minus119902119904)

Lemma 1 Let 119862 be a 120582-constacyclic code over F1199022 with1199001199031198891199022(120582) = 119903 If 119862 is a Hermitian self-orthogonal code then119903 | (119902 + 1)

Proof The proof is similar to [6 Proposition 23]

Theorem 2 Nontrivial Hermitian self-orthogonal 120582-consta-cyclic codes of length 119899 over F1199022 exist if and only if 119862119904 = 119862minus119902119904for some 119904 isin 119874119903119899(1)

Proof Let 119862 be a nontrivial Hermitian self-orthogonal 120582-constacyclic code of length 119899 over F1199022 with defining set 119879Then there exists 119904 isin 119874119903119899(1) such that 119904 = 0 and 119904 notin 119879













(1)

1= Λ 1 Λ

(0)

1 Observe that

1198790 = ⋃119904isinΛ(0)

1

119862119904


119874715 (1)

= 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

(6)

1198790 = 15 Let 119879 = 1 15 22 36 43 50 64 99 then 119879perp119867=



(1)

1|2


prod119904isinΛ1


(0)

1


1 the pairs (120575(119904) 120575(minus119902119904))


|Λ(1)

1|2




119886119909 equiv 1 (mod 119903) (7)





119896=0119860(119896)

119887 where 119860(119896)

119887= 119896119901119903 + 119887 (119896119901 +



mod119901 each 119860(119896)119887







1198730 = sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 119889119894119904119905119894119899119888119905

119889

119901119894119901119895119901119896


11990111199012 119901119898

(8)




Now

119873 = 119889 minus

sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 distinct

119889

119901119894119901119895119901119896


11990111199012 sdot sdot sdot 119901119898

= 119889(1 minus

1

1199011

)(1 minus

1

1199012


1

119901119898

)

(9)


119898+119897+119896

119894=1(1 minus


119904

= 119904 isin

Λ 1 gcd(119903119899 119904) = 119898119904

Theorem 11 Consider

10038161003816100381610038161003816Λ 1119898

119904

10038161003816100381610038161003816=

120601 (119903119889119904)

119900119903119889119903119889119904

(1199022) 120601 (119903)

119894119891 119892119888119889 (119898119904 119903) = 1

0 119894119891 119892119888119889 (119898119904 119903) = 1

(10)

where 119889119904 = 119899119898119904


119904

| =


1015840Also gcd(119889119904 119904






1015840equiv 1(mod 119903)


1003816100381610038161003816119862119904

1003816100381610038161003816= ord119903119899119898

119904

(1199022) = ord119903119889

119904

(1199022) (11)


| = 120601(119903119889119904)ord119903119889119904

(1199022)120601(119903) whenever

gcd(119898119904 119903) = 1

Define

120594 (119889 119902) =


1 otherwise(12)


Theorem 12 One has10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816= sum

119889|119899

119892119888119889(119899119889119903)=1

120594 (119903119889 119902) 120601 (119903119889)

120601 (119903) 119900119903119889rd (1199022)

(13)






10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816=

120594 (36 5) 120601 (36)

120601 (3) ord36 (25)=

12

3 times 2

= 2 (14)



Proof Write 11989811198982 = 2119886+119887prod119898

119894=1119901119890119894





119902119896equiv minus1 (mod11989811198982) (15)

where 119896 = 119897119888119898119896119901119894119890119894

119898

119894=1is odd






119896119901equiv minus1(mod119901)











1199022




By Lemma 1 119903 | (119902 + 1) Write 119902 + 1 = 1199031199040


1198970 =

minuslfloor

1199040 minus 1

2

rfloor 119894119891 119899 119894119904 119890V119890119899

minus lfloor

1199040 minus 2

2

rfloor 119894119891 119899 119894119904 119900119889119889

(16)



119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) (17)



1199040 + 119894 + 119895 = 119899 equiv 0 (mod 119899) (18)


so that

0 lt 1199040 + 119894 + 119895 lt 119899 (19)




119873 =

119899 (119899 minus 2)

8

119894119891 119904 119894119904 119890V119890119899 119899 119890V119890119899

119899 (119899 + 2)

8

119894119891 119904 119894119904 119900119889119889 119899 119890V119890119899(119899 minus 1) (119899 + 1)

8

119894119891 119904 119894119904 119890V119890119899 119899 119900119889119889

(119899 minus 3) (119899 minus 1)

8

119894119891 119904 119894119904 119900119889119889 119899 119900119889119889

(20)


119895=1119895 = (1198980 minus 1198970 + 1)(1198980 minus





119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109




References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















(1)

1= Λ 1 Λ

(0)

1 Observe that

1198790 = ⋃119904isinΛ(0)

1

119862119904


119874715 (1)

= 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

(6)

1198790 = 15 Let 119879 = 1 15 22 36 43 50 64 99 then 119879perp119867=



(1)

1|2


prod119904isinΛ1


(0)

1


1 the pairs (120575(119904) 120575(minus119902119904))


|Λ(1)

1|2




119886119909 equiv 1 (mod 119903) (7)





119896=0119860(119896)

119887 where 119860(119896)

119887= 119896119901119903 + 119887 (119896119901 +



mod119901 each 119860(119896)119887







1198730 = sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 119889119894119904119905119894119899119888119905

119889

119901119894119901119895119901119896


11990111199012 119901119898

(8)




Now

119873 = 119889 minus

sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 distinct

119889

119901119894119901119895119901119896


11990111199012 sdot sdot sdot 119901119898

= 119889(1 minus

1

1199011

)(1 minus

1

1199012


1

119901119898

)

(9)


119898+119897+119896

119894=1(1 minus


119904

= 119904 isin

Λ 1 gcd(119903119899 119904) = 119898119904

Theorem 11 Consider

10038161003816100381610038161003816Λ 1119898

119904

10038161003816100381610038161003816=

120601 (119903119889119904)

119900119903119889119903119889119904

(1199022) 120601 (119903)

119894119891 119892119888119889 (119898119904 119903) = 1

0 119894119891 119892119888119889 (119898119904 119903) = 1

(10)

where 119889119904 = 119899119898119904


119904

| =


1015840Also gcd(119889119904 119904






1015840equiv 1(mod 119903)


1003816100381610038161003816119862119904

1003816100381610038161003816= ord119903119899119898

119904

(1199022) = ord119903119889

119904

(1199022) (11)


| = 120601(119903119889119904)ord119903119889119904

(1199022)120601(119903) whenever

gcd(119898119904 119903) = 1

Define

120594 (119889 119902) =


1 otherwise(12)


Theorem 12 One has10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816= sum

119889|119899

119892119888119889(119899119889119903)=1

120594 (119903119889 119902) 120601 (119903119889)

120601 (119903) 119900119903119889rd (1199022)

(13)






10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816=

120594 (36 5) 120601 (36)

120601 (3) ord36 (25)=

12

3 times 2

= 2 (14)



Proof Write 11989811198982 = 2119886+119887prod119898

119894=1119901119890119894





119902119896equiv minus1 (mod11989811198982) (15)

where 119896 = 119897119888119898119896119901119894119890119894

119898

119894=1is odd






119896119901equiv minus1(mod119901)











1199022




By Lemma 1 119903 | (119902 + 1) Write 119902 + 1 = 1199031199040


1198970 =

minuslfloor

1199040 minus 1

2

rfloor 119894119891 119899 119894119904 119890V119890119899

minus lfloor

1199040 minus 2

2

rfloor 119894119891 119899 119894119904 119900119889119889

(16)



119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) (17)



1199040 + 119894 + 119895 = 119899 equiv 0 (mod 119899) (18)


so that

0 lt 1199040 + 119894 + 119895 lt 119899 (19)




119873 =

119899 (119899 minus 2)

8

119894119891 119904 119894119904 119890V119890119899 119899 119890V119890119899

119899 (119899 + 2)

8

119894119891 119904 119894119904 119900119889119889 119899 119890V119890119899(119899 minus 1) (119899 + 1)

8

119894119891 119904 119894119904 119890V119890119899 119899 119900119889119889

(119899 minus 3) (119899 minus 1)

8

119894119891 119904 119894119904 119900119889119889 119899 119900119889119889

(20)


119895=1119895 = (1198980 minus 1198970 + 1)(1198980 minus





119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109




References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Now

119873 = 119889 minus

sum

1le119894le119898

119889

119901119894

minus sum

1le119894 119895le119898

119894 =119895

119889

119901119894119901119895

+ sum

1le119894119895119896le119898

119894119895119896 distinct

119889

119901119894119901119895119901119896


11990111199012 sdot sdot sdot 119901119898

= 119889(1 minus

1

1199011

)(1 minus

1

1199012


1

119901119898

)

(9)


119898+119897+119896

119894=1(1 minus


119904

= 119904 isin

Λ 1 gcd(119903119899 119904) = 119898119904

Theorem 11 Consider

10038161003816100381610038161003816Λ 1119898

119904

10038161003816100381610038161003816=

120601 (119903119889119904)

119900119903119889119903119889119904

(1199022) 120601 (119903)

119894119891 119892119888119889 (119898119904 119903) = 1

0 119894119891 119892119888119889 (119898119904 119903) = 1

(10)

where 119889119904 = 119899119898119904


119904

| =


1015840Also gcd(119889119904 119904






1015840equiv 1(mod 119903)


1003816100381610038161003816119862119904

1003816100381610038161003816= ord119903119899119898

119904

(1199022) = ord119903119889

119904

(1199022) (11)


| = 120601(119903119889119904)ord119903119889119904

(1199022)120601(119903) whenever

gcd(119898119904 119903) = 1

Define

120594 (119889 119902) =


1 otherwise(12)


Theorem 12 One has10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816= sum

119889|119899

119892119888119889(119899119889119903)=1

120594 (119903119889 119902) 120601 (119903119889)

120601 (119903) 119900119903119889rd (1199022)

(13)






10038161003816100381610038161003816Λ(1)

1

10038161003816100381610038161003816=

120594 (36 5) 120601 (36)

120601 (3) ord36 (25)=

12

3 times 2

= 2 (14)



Proof Write 11989811198982 = 2119886+119887prod119898

119894=1119901119890119894





119902119896equiv minus1 (mod11989811198982) (15)

where 119896 = 119897119888119898119896119901119894119890119894

119898

119894=1is odd






119896119901equiv minus1(mod119901)











1199022




By Lemma 1 119903 | (119902 + 1) Write 119902 + 1 = 1199031199040


1198970 =

minuslfloor

1199040 minus 1

2

rfloor 119894119891 119899 119894119904 119890V119890119899

minus lfloor

1199040 minus 2

2

rfloor 119894119891 119899 119894119904 119900119889119889

(16)



119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) (17)



1199040 + 119894 + 119895 = 119899 equiv 0 (mod 119899) (18)


so that

0 lt 1199040 + 119894 + 119895 lt 119899 (19)




119873 =

119899 (119899 minus 2)

8

119894119891 119904 119894119904 119890V119890119899 119899 119890V119890119899

119899 (119899 + 2)

8

119894119891 119904 119894119904 119900119889119889 119899 119890V119890119899(119899 minus 1) (119899 + 1)

8

119894119891 119904 119894119904 119890V119890119899 119899 119900119889119889

(119899 minus 3) (119899 minus 1)

8

119894119891 119904 119894119904 119900119889119889 119899 119900119889119889

(20)


119895=1119895 = (1198980 minus 1198970 + 1)(1198980 minus





119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109




References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















1199022




By Lemma 1 119903 | (119902 + 1) Write 119902 + 1 = 1199031199040


1198970 =

minuslfloor

1199040 minus 1

2

rfloor 119894119891 119899 119894119904 119890V119890119899

minus lfloor

1199040 minus 2

2

rfloor 119894119891 119899 119894119904 119900119889119889

(16)



119879119897119898 = 119894119903 + 1 119897 le 119894 le 119898 (mod 119903119899) (17)



1199040 + 119894 + 119895 = 119899 equiv 0 (mod 119899) (18)


so that

0 lt 1199040 + 119894 + 119895 lt 119899 (19)




119873 =

119899 (119899 minus 2)

8

119894119891 119904 119894119904 119890V119890119899 119899 119890V119890119899

119899 (119899 + 2)

8

119894119891 119904 119894119904 119900119889119889 119899 119890V119890119899(119899 minus 1) (119899 + 1)

8

119894119891 119904 119894119904 119890V119890119899 119899 119900119889119889

(119899 minus 3) (119899 minus 1)

8

119894119891 119904 119894119904 119900119889119889 119899 119900119889119889

(20)


119895=1119895 = (1198980 minus 1198970 + 1)(1198980 minus





119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109




References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902 119899 119903 119873 119896 119889 119879

5 4 1 1 1 4 0 1 2

5 4 2 32 3 3 5

1 4 1 3 5

1 4 3 5 7

5 4 3 1 1 4 4 7 10

5 4 6 32 3 13 19

1 4 7 13 19

1 4 1 13 19

7 3 1 1 1 3 0 2

7 3 2 1 1 3 1 3

7 3 4 1 1 3 5 9

7 6 1 32 5 0 1 2 3

1 6 0 1 2 3 4

1 6 0 1 2 3 5

7 6 2 32 5 3 5 7 9

1 6 1 3 5 7 9

1 6 3 5 7 9 11

7 6 4 32 5 9 13 17 21

1 6 1 9 13 17 21

1 6 5 9 13 17 21

7 6 8 6

3 4 25 33 41

2 5 17 25 33 41

2 5 1 25 33 41

1 6 9 17 25 33 41

1 6 1 17 25 33 41

1 6 1 9 25 33 41

11 5 1 32 4 0 3 4

1 5 0 2 3 4

1 5 0 1 3 4

11 5 2 32 4 1 3 5

1 5 1 3 5 9

1 5 1 3 5 7

11 5 3 32 4 4 7 10

1 5 1 4 7 10

1 5 4 7 10 13

11 5 4 1 1 5 5 9 13 17

11 5 6 32 4 13 19 25

1 5 7 13 19 25

1 5 1 13 19 25

11 5 12 1 1 5 25 37 49 1

11 10 1 10

4 7 0 1 2 3 4 5

3 8 0 1 2 3 4 5 9

3 8 0 1 2 3 4 5 6

2 9 0 1 2 3 4 5 8 9

2 9 0 1 2 3 4 5 6 9

2 9 0 1 2 3 4 5 6 7

1 10 0 1 2 3 4 5 7 8 9

1 10 0 1 2 3 4 5 6 8 9

1 10 0 1 2 3 4 5 6 7 9

1 10 0 1 2 3 4 5 6 7 8

11 10 2 10

4 7 5 7 9 11 13 15

3 8 3 5 7 9 11 13 15

3 8 5 7 9 11 13 15 17

2 9 1 3 5 7 9 11 13 15

2 9 3 5 7 9 11 13 15 17

2 9 5 7 9 11 13 15 17 19

1 10 1 3 5 7 9 11 13 15 19

1 10 1 3 5 7 9 11 13 15 17

1 10 3 5 7 9 11 13 15 17 19

1 10 1 5 7 9 11 13 15 17 19


119902 119899 119903 119873 119896 119889 119879

11 10 3 10

4 7 10 13 16 19 22 25

3 8 7 10 13 16 19 22 25

3 8 10 13 16 19 22 25 28

2 9 4 7 10 13 16 19 22 25

2 9 7 10 13 16 19 22 25 28

2 9 1 10 13 16 19 22 25 28

1 10 1 4 7 10 13 16 19 22 25

1 10 4 7 10 13 16 19 22 25 28

1 10 1 7 10 13 16 19 22 25 28

1 10 1 4 10 13 16 19 22 25 28

11 10 4 15

5 6 17 21 25 29 33

4 7 13 17 21 25 29 33

4 7 17 21 25 29 33 37

3 8 9 13 17 21 25 29 33

3 8 13 17 21 25 29 33 37

3 8 1 17 21 25 29 33 37

2 9 5 9 13 17 21 25 29 33

2 9 9 13 17 21 25 29 33 37

2 9 1 13 17 21 25 29 33 37

2 9 1 5 17 21 25 29 33 37

1 10 1 5 9 13 17 21 25 29 33

1 10 5 9 13 17 21 25 29 33 37

1 10 1 9 1317 21 25 29 33 37

1 10 1 5 1317 21 25 29 33 37

1 10 1 5 9 17 21 25 29 33 37

11 10 6 10

4 7 25 31 37 43 49 55

3 8 19 25 31 37 43 49 55

3 8 1 25 31 37 43 49 55

2 9 13 19 25 31 37 43 49 55

2 9 1 19 25 31 37 43 49 55

2 9 1 7 25 31 37 43 49 55

1 10 7 13 19 25 31 37 43 49 55

1 10 1 13 19 25 31 37 43 49 55

1 10 1 7 19 25 31 37 43 49 55

1 10 1 7 13 25 31 37 43 49 55

11 10 12 15

5 6 61 73 85 97 109

4 7 49 61 73 85 97 109

4 7 1 61 73 85 97 109

3 8 37 49 61 73 85 97 109

3 8 1 49 61 73 85 97 109

3 8 1 13 61 73 85 97 109

2 9 25 37 49 61 73 85 97 109

2 9 1 37 49 61 73 85 97 109

2 9 1 13 49 61 73 85 97 109

2 9 1 13 25 61 73 85 97 109

1 10 13 25 37 49 61 73 85 97 109

1 10 1 25 37 49 61 73 85 97 109

1 10 1 13 37 49 61 73 85 97 109

1 10 1 13 25 49 61 73 85 97 109

1 10 1 13 25 3761 73 85 97 109




References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields