\(\mathbb Z_{8}\)-Kerdock codes and pseudorandom binary sequences
type: Item
instance of: scholarly article
title: \(\mathbb Z_{8}\)-Kerdock codes and pseudorandom binary sequences
zbMATH Open document ID: 1059.94013
author: Jyrki Lahtonen
author: San Ling
author: Patrick Solé
author: Dmitrii Zinoviev
published in: Journal of Complexity
publication date: 2004-08-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 94A55
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 94B60
zbMATH DE Number: 2083572
zbMATH Keywords: Generalized Kerdock code
zbMATH Keywords: Most significant bit map
zbMATH Keywords: Boolean functions
zbMATH Keywords: Correlation
zbMATH Keywords: Nonlinearity
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
full work available at URL: j.jco.2003.08.014
OpenAlex ID: W2110585483
cites work: Q4954419
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DOI: 10.1016/J.JCO.2003.08.014
description: scientific article; zbMATH DE number 2083572