On correlation distribution of Niho-type decimation $d=3(p^m-1)+1$

TL;DR

This paper determines the cross-correlation distribution of a specific Niho-type decimation over finite fields for primes p ≥ 5, extending previous results for smaller primes and employing advanced algebraic geometry and coding theory techniques.

Contribution

It generalizes the cross-correlation distribution computation to primes p ≥ 5, using novel transformations and algebraic geometry methods previously not applied to this problem.

Findings

Cross-correlation distribution computed for p ≥ 5

Transformation to codeword counting in Melas codes

Use of elliptic curves for p ≥ 7

Abstract

The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation $d=3(p^m-1)+1$ over $\mathrm{GF}(p^{2m})$ for any prime $p \ge 5$. Previously this problem was solved by Xia et al. only for $p=2$ and $p=3$ in a series of papers. The main difficulty of this problem for $p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in $p$-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in $p$-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When $p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.