A New Method to Compute the 2-adic Complexity of Binary Sequences

TL;DR

This paper introduces a novel method for calculating the 2-adic complexity of binary sequences, demonstrating that sequences with ideal autocorrelation have maximum complexity and extending the approach to other finite fields.

Contribution

A new method for computing 2-adic complexity is proposed, enabling uniform analysis of sequences with ideal autocorrelation and extending to sequences over different finite fields.

Findings

Sequences with ideal 2-level autocorrelation have maximum 2-adic complexity equal to their period.

The method effectively computes 2-adic complexities of Legendre and Ding-Helleseth-Lam sequences.

The approach can also determine linear complexity over other finite fields.

Abstract

In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are uniformly determined. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum. Moreover, 2-adic complexities of two classes of optimal autocorrelation sequences with period $N\equiv1\mod4$, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated. Besides, this method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.