The merit factor of binary arrays derived from the quadratic character

We calculate the asymptotic merit factor, under all cyclic rotations of rows and columns, of two families of binary two-dimensional arrays derived from the quadratic character. The arrays in these families have size p x q, where p and q are not necessarily distinct odd primes, and can be considered as two-dimensional generalisations of a Legendre sequence. The asymptotic values of the merit factor of the two families are generally different, although the maximum asymptotic merit factor, taken over all cyclic rotations of rows and columns, equals 36/13 for both families. These are the first non-trivial theoretical results for the asymptotic merit factor of families of truly two-dimensional binary arrays.

[1]  Michael Schroeder.  Number theory in science and communication : With applications in cryptog-raphy , 1997 .

[2]  J. Storer,et al.  On binary sequences , 1961 .

[3]  M. R. Schroeder,et al.  Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity (Springer Series in Information Sciences) , 1986 .

[4]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[5]  Michael J. Mossinghoff.  Wieferich pairs and Barker sequences , 2009, Des. Codes Cryptogr..

[6]  Peter B. Borwein,et al.  Binary sequences with merit factor greater than 6.34 , 2004, IEEE Transactions on Information Theory.

[7]  M. Schroeder.  Number Theory in Science and Communication , 1984 .

[8]  Binary sequences with good correlation properties , 1975 .

[9]  James A. Davis,et al.  Proof of the Barker Array Conjecture , 2007 .

[10]  Jack K. Wolf,et al.  On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties , 1967, Inf. Control..

[11]  W. Marsden.  I and J , 2012 .

[12]  Matthew G. Parker,et al.  Two binary sequence families with large merit factor , 2009, Adv. Math. Commun..

[13]  Hans D. Schotten,et al.  Quadratic Residue Arrays , 1993 .

[14]  Robert A. Scholtz,et al.  On the nonexistence of Barker arrays and related matters , 1989, IEEE Trans. Inf. Theory.

[15]  Tom Høholdt,et al.  The merit factor of binary sequences related to difference sets , 1991, IEEE Trans. Inf. Theory.

[16]  Leopold Bömer,et al.  Optimizing the aperiodic merit factor of binary arrays , 1993, Signal Process..

[17]  Jonathan Jedwab,et al.  The L_4 norm of Littlewood polynomials derived from the Jacobi symbol , 2011, ArXiv.

[18]  Tom Høholdt,et al.  Aperiodic correlations and the merit factor of a class of binary sequences , 1985, IEEE Trans. Inf. Theory.

[19]  Jonathan Jedwab,et al.  The merit factor of binary sequence families constructed from m-sequences , 2009 .

[20]  D. V. Sarwate,et al.  Mean-square correlation of shift-register sequences , 1984 .

[21]  M.R. Schroeder,et al.  Number theory , 1989, IEEE Potentials.

[22]  T. Aaron Gulliver,et al.  The multivariate merit factor of a Boolean function , 2005, IEEE Information Theory Workshop, 2005..

[23]  J. Littlewood.  Some problems in real and complex analysis , 1968 .

[24]  Tom Høholdt,et al.  Determination of the merit factor of Legendre sequences , 1988, IEEE Trans. Inf. Theory.

[25]  Tom Høholdt,et al.  Binary Sequences with Good Correlation Properties , 1987, AAECC.

[26]  Imants D. Svalbe,et al.  Algebraic construction of a new class of quasi-orthogonal arrays for steganography , 1999, Electronic Imaging.

[27]  Jonathan Jedwab,et al.  A Survey of the Merit Factor Problem for Binary Sequences , 2004, SETA.