\(\mathbb Z_{8}\)-Kerdock codes and pseudorandom binary sequences
There is quite a bit of literature on sequences with pseudorandom properties. The desired properties split up into statistical properties like low non-central autocorrelation and low overall cross correlation, and cryptographical properties like large linear span and high nonlinearity. In this paper, using Kerdock codes over \(\mathbb{Z}\text{\,mod\,}8\), a family of binary sequences of length \(T= 2^m- 1\), having reasonable correlation properties and a linear span of the order \(O(m^4)\), is constructed. The size of the family is \(T^2/4\). The somewhat larger order of the linear span is compensated by, compared to another family constructed from Galois rings of characteristic 4, somewhat worse bounds for cross correlation and nonlinearity respectively being \(3(2+ \sqrt{2})(\sqrt{T+ 1})\) and \(3\sqrt{2+ \sqrt{2}}\sqrt{T+ 1}\). The proof of the main result starts with showing that the most significant bit mapping and the Lee weight can be expressed in terms of group characters. Then some elementary character theory can be used combined with a Galois ring analogue of the Weil inequality. In the last section of the paper the results are generalised to \(\mathbb{Z}\text{\,mod\,}2^l\), but the problem of determining the nonlinearity and the linear span for this generalisation is left open.
scientific article; zbMATH DE number 2083572