Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q = pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn - 1 for q > 2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qn - 1 is difference-balanced and has the "cyclic" array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qn - 1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q = p.

[1]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[2]  Guang Gong,et al.  New nonbinary sequences with ideal two-level autocorrelation function , 2002, Proceedings IEEE International Symposium on Information Theory,.

[3]  L. D. Baumert.  Cyclic Difference Sets , 1971 .

[4]  Hong-Yeop Song,et al.  Expanding generalized Hadamard matrices over Gm by substituting several generalized Hadamard matrices over G , 2001, Journal of Communications and Networks.

[5]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[6]  R. McEliece.  Finite field for scientists and engineers , 1987 .

[7]  Guang Gong,et al.  Q-ary cascaded GMW sequences , 1996, IEEE Trans. Inf. Theory.

[8]  Solomon W. Golomb,et al.  Shift Register Sequences , 1981 .

[9]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[10]  Marshall Hall,et al.  A survey of difference sets , 1956 .

[11]  J. Singer.  A theorem in finite projective geometry and some applications to number theory , 1938 .

[12]  S. Chowla,et al.  On Difference Sets. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Tor Helleseth,et al.  New nonbinary sequences with ideal two-level autocorrelation , 2002, IEEE Trans. Inf. Theory.

[14]  Jong-Seon No,et al.  New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions , 2004, Des. Codes Cryptogr..

[15]  R. McEliece.  Finite Fields for Computer Scientists and Engineers , 1986 .

[16]  Andrew Klapper,et al.  D-form Sequences: Families of Sequences with Low Correlation Values and Large Linear Spans , 1995, IEEE Trans. Inf. Theory.

[17]  Rudolf Lide,et al.  Finite fields , 1983 .

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

[19]  Mark Goresky,et al.  Cascaded GMW sequences , 1993, IEEE Trans. Inf. Theory.