1846 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004

Proof: Again, let = �(1+2�) 2 R, where � 2 T is a generatorof the Teichmüller set and � 2 R�. As we have ct = MSB(Tr(� t)),and by (2), we obtain that (�1)c is equal to

�(Tr(� t)) =

2 �1

j=0

�j j(Tr(� t)) =

2 �1

j=0

�j�j( t)):

Changing the order of summation, we obtain that

�(� ) =

2 �1

j =0

2 �1

j =0

�j �j

N�1

t=0

�( t):

Here, � = �(j1 + j2 � ) and j1 + j2

� 6= 0 as � =2 2 . ApplyingCorollary 7.4 of [9] (for l � 4), we have

2 �1

j =0

2 �1

j =0

j�j �j j =2 �1

j=0

j�j j2

� 2l

�ln(2) + 1

2

: (8)

Applying Lemma 3.2, we have

N�1

j=0

�( j) =

2 �1

j=0 x2T

�(1+2�) (x)

where 0 6= �(1 + 2�)j 2 R so that each sum over x can be estimatedusing Lemma 3.1. Thus, we have

N�1

j=0

�( j) � 2l�1[(2l�1 � 1)

p2m + 1]: (9)

Combining (8) with (9) the lemma follows.

The estimate of [4, Theorem 3] is of the same shape but with a C0l

of the order (for large l) of 23l.

IV. CONCLUSION

In this note, we improve on the results of [4] that were also usingGalois ring character sums bounds of [7]. This improvement is only dueto the use of the Vinogradov method for incomplete character sums [3],already employed to control the aperiodic correlation of some classicalfamilies of low correlation sequences [8]. The approach of [4] was todivide each sequence into 2l�1 subsequences. Therefore, a factor oforder 2l appeared in their bounds.

ACKNOWLEDGMENT

The authors thank the referees for helpful suggestions and Prof.Z.-D. Dai for sending them [2].

REFERENCES

[1] Z.-D. Dai, “Binary sequences derived from ML-sequences over rings I:Period and minimal polynomial,” J. Cryptol., vol. 5, pp. 193–507, 1992.

[2] Z.-D. Dai, Y. Dingfeng, W. Ping, and F. Genxi, “Distribution of r-pat-terns in the highest level of p-adic sequences over Galois rings,” in Proc.Golomb Symp., Univ. Southern California, Los Angeles, CA, 2002.

[3] H. Davenport, Multiplicative Number Theory. New York: Springer-Verlag, 2000, vol. GTM 74.

[4] S. Fan and W. Han, “Random properties of the highest level sequencesof primitive sequences over ,” IEEE Trans. Inform. Theory, vol. 49,pp. 1553–1557, June 2003.

[5] T. Helleseth and P. V. Kumar, “Sequences with low correlation,”in Handbook of Coding Theory, V. S. Pless and W. C. Huffman,Eds. Amsterdam, The Netherlands: North-Holland, 1998, vol. II, pp.1765–1853.

[6] P. V. Kumar and T. Helleseth, “An expansion of the coordinates of thetrace function over Galois rings,” AAECC, vol. 8, pp. 353–361, 1997.

[7] P. V. Kumar, T. Helleseth, and A. R. Calderbank, “An upper bound forWeil exponential sums over Galois rings and applications,” IEEE Trans.Inform. Theory, vol. 41, pp. 456–468, May 1995.

[8] J. Lahtonen, “On the odd and the aperiodic correlation properties of thebinary Kasami sequences,” IEEE Trans. Inform. Theory, vol. 41, pp.1506–1508, Sept. 1995.

[9] J. Lahtonen, S. Ling, P. Solé, and D. Zinoviev, “ -Kerdock codes andpseudo-random binary sequences,” J. Complexity, to be published.

[10] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North-Holland, 1977.

Symmetric Golomb Squares

James B. Shearer

Abstract—Inspection of optimum Golomb squares found by exhaustivecomputer search shows some are symmetric with respect to diagonal reflec-tion. This suggests searching for such symmetric Golomb squares might bean efficient way to find good squares. In this correspondence, we find thebest symmetric Golomb squares for 22. Some appear tobe the best known.

Index Terms—Backtrack, Golomb, search, square, symmetric.

I. AMS SUBJECT CLASSIFICATION. 05B99

In [1], Robinson defined a Golomb rectangle as anN �M array ofones and zeros such that the two-dimensional autocorrelation has threevalues: 0, 1, and K , whereK is the number of ones in the array. Thismeans that the positions of the ones in any nonzero integral transla-tion of the rectangle will overlap with the positions of the ones in theoriginal position of the rectangle in at most one place. Equivalently, thedifferences between the positions of every pair of ones in the rectangle,considered as vectors, are distinct. See also [2]. Let G(N;M) be themaximum number of ones that can be present in an N �M Golombrectangle. For example, G(2; 2) = 3. A Golomb square is a Golombrectangle withN =M . An optimum Golomb square is one containingG(N;N) ones.

Optimum Golomb squares can be found by exhaustive backtracksearch. However, the search time increases rapidly with N . In [3], wefound G(N;N) for N � 10 by exhaustive computer search. Usinga similar program, we extended this work and found all distinct op-timum Golomb squares for N � 12. Inspection shows some of theseoptimum Golomb squares are symmetric with respect to diagonal re-flection. These results are summarized in Table I. For each N we giveG(N;N), the number of distinct optimum squares, the number of dis-tinct diagonally symmetric squares, and the search time in seconds (onan IBM RS/6000 model 7043-140 which uses a 332-MHz 604e chip).Note that except for N = 7 some of the optimum Golomb squaresare diagonally symmetric. Note also that the computation time is in-creasing rapidly with N . Since it is much faster to search for diag-onally symmetric Golomb squares than arbitrary Golomb squares thissuggests searching for diagonally symmetric Golomb squares might bean efficient heuristic for finding good Golomb squares.

Manuscript received October 9, 2003; revised February 27, 2004.The author is with the IBM Research Division, T. J. Watson Research Center,

P. O. Box 218, Yorktown Heights, NY 10598 USA (e-mail: jbs@watson.ibm.com).

Communicated by K. G. Paterson, Associate Editor for Sequences.Digital Object Identifier 10.1109/TIT.2004.831839

0018-9448/04$20.00 © 2004 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 1847

TABLE IARBITRARY SEARCH

Fig. 1. Arbitrary search.

Fig. 2. Symmetric search.

Wemodified our program to exhaustively search for diagonally sym-metric Golomb squares. As described in [3] (see also [4]) searching forN � N Golomb squares is equivalent to searching for Golomb rulerswith marks chosen from the set

fi+ (j � 1)� (2�N � 1)j1 � i; j � Ng:

To search for symmetric Golomb squares it is more expedient to searchfor Golomb rulers with marks chosen from the set

f1+(i�1)� (2�N�2) + (j�1)� (2�N�1)j1� i; j�Ng:

Figs. 1 and 2 show how these numbering schemes differ when search-ing for 5 � 5Golomb squares. However, in either case each differencecorresponds to a unique vector offset. In order to force the squares tobe symmetric, each element in the lower left part of the square is pairedwith its diagonal reflection in the upper right part of the square. (i; j) ispaired with (j; i). The backtrack search program places and removesthese pairs as an unit. Table II shows the results from the modifiedprogram. S(N) is the maximum number of ones in a symmetricN�N

Golomb square. Clearly, G(N;N) � S(N).We can see that searching for symmetric Golomb squares is a more

efficient way of finding good Golomb squares than searching for ar-bitrary Golomb squares. For example, when N = 12 an exhaustivesearch for arbitrary Golomb squares with 17 ones takes 335680.88 s tofind 2635 squares or 127 s per square. But an exhaustive search for sym-metric Golomb squares takes just 0.33 s to find 15 squares or 0.022 sper square.

The bounds for G(N;N) for 13 � N � 22 established by thesesearches are the best known to the author. Robinson [5] used a geneticsearch algorithm to find good Golomb squares. Our results are as goodfor 11 � N � 14 and one better for N = 15; 20. In Fig. 3, we givethe unique symmetric 22 � 22 Golomb square containing 29 ones.

One of the referees pointed out that examples of Golomb squareswith N = 20 and 26 ones were already known (see [6]).

TABLE IISYMMETRIC SEARCH

Fig. 3. S(22) = 29.

Examples of symmetric Golomb squares achieving the results inthis correspondence can be obtained from the author on request andshould eventually appear on the author’s website (http://www.research.ibm.com/people/s/shearer/home.html).

REFERENCES

[1] J. P. Robinson, “Golomb rectangles,” IEEE Trans. Inform. Theory, vol.IT-31, pp. 781–787, Nov. 1985.

[2] S. W. Golomb and M. Taylor, “Two-dimensional synchronization pat-terns for minimal ambiguity,” IEEE Trans. Inform. Theory, vol. IT-28,pp. 600–604, July 1982.

[3] J. B. Shearer, “Some new optimum Golomb rectangles,” Electron. J.Comb., vol. 2, no. #R12, 1995. [Online].

[4] J. P. Robinson, “Golomb rectangles as folded rulers,” IEEE Trans. In-form. Theory, vol. 43, pp. 290–293, Jan. 1997.

[5] , “Genetic search for Golomb arrays,” IEEE Trans. Inform. Theory,vol. 46, pp. 1170–1173, May 2000.

[6] L. E. Kopilovich and L. G. Sodin, Multielement System Design in As-tronomy and Radio Science. Norwell, MA: Kluwer Academic, 2001.