A note on the five valued conjectures of Johansen and Helleseth and zeta functions

2013/09/23 by Xiaogang Liu, Michael Harrison, Liu, Xiaogang +3
Computer Science · Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.1309.5674

openalex publication_date 2013/09/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For the complete five-valued cross-correlation distribution between two m-sequences s_t and s_dt of period 2^m-1 that differ by the decimation d=2^2k+1\over 2^k+1 where m is odd and gcd(k,m)=1, Johansen and Hellseth expressed it in terms of some exponential sums. And two conjectures are presented that are of interest in their own right. In this correspondence we study these conjectures for the particular case where k=3, and the cases k=1,2 can also be analyzed in a similar process. When k>3, the degrees of the relevant polynomials will become higher. Here the multiplicity of the biggest absolute value of the cross-correlation is no more than one-sixth of the multiplicity corresponding the smallest absolute value.

Citations

• Some results about the cross-correlation function between two maximal linear sequences
• A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

Related

• Period-Different m-Sequences With At Most A Four-Valued Cross Correlation
• Some New Results on the Cross Correlation of m-sequences
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• m-Sequences of Different Lengths with Four-Valued Cross Correlation
• Proof of a Conjecture of Helleseth: Maximal Linear Recursive Sequences of Period 2²ⁿ-1 Never Have Three-Valued Cross-Correlation