A Family of Binary Sequences with 4-Valued Optimal Out-of-Phase Correlation and Large Linear Span

A family of binary sequences is presented and proved to have optimal correlation property and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. An explicit lower bound expression on the linear span of sequences in the family is given. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences. A class of ideal autocorrelation sequences is also constructed and proved to have large linear span.

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

[2]  P. Vijay Kumar,et al.  A new family of binary pseudorandom sequences having optimal periodic correlation properties and large linear span , 1989, IEEE Trans. Inf. Theory.

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

[4]  Guang Gong,et al.  New designs for signal sets with low cross correlation, balance property, and largelinear span: GF(p) case , 2002, IEEE Trans. Inf. Theory.

[5]  T. Kasami.  WEIGHT DISTRIBUTION OF BOSE-CHAUDHURI-HOCQUENGHEM CODES. , 1966 .

[6]  Jong-Seon No,et al.  P-ary Unified Sequences: P-ary Extended D-form Sequences with the Ideal Autocorrelation Property , 2002, IEEE Trans. Inf. Theory.

[7]  Tor Helleseth,et al.  Correlation of m-Sequences and Related Topics , 1998, SETA.

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

[9]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[10]  Edwin L. Key,et al.  An analysis of the structure and complexity of nonlinear binary sequence generators , 1976, IEEE Trans. Inf. Theory.

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

[12]  R. Gold,et al.  Optimal binary sequences for spread spectrum multiplexing (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[13]  Guang Gong,et al.  Binary Sequences with Two-Level Autocorrelation , 1999, IEEE Trans. Inf. Theory.

[14]  Jong-Seon No,et al.  Linear span of extended sequences and cascaded GMW sequences , 1999, IEEE Trans. Inf. Theory.

[15]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[16]  T. Kasami.  WEIGHT DISTRIBUTION FORMULA FOR SOME CLASS OF CYCLIC CODES , 1966 .

[17]  Marvin K. Simon,et al.  Spread Spectrum Communications Handbook , 1994 .

[18]  Hong-Yeop Song,et al.  New construction for families of binary sequences with optimal correlation properties , 1997, IEEE Trans. Inf. Theory.

[19]  Hong-Yeop Song,et al.  Trace representation of Legendre sequences of Mersenne prime period , 1996, IEEE Trans. Inf. Theory.

[20]  Robert A. Scholtz,et al.  Bent-function sequences , 1982, IEEE Trans. Inf. Theory.

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

[22]  P. Vijay Kumar,et al.  Bounds on the linear span of bent sequences , 1983, IEEE Trans. Inf. Theory.