An Assmus-Mattson-type approach for identifying 3-designs from linear codes over \(\mathbb{Z}_4\)
type: Item
instance of: scholarly article
title: An Assmus-Mattson-type approach for identifying 3-designs from linear codes over \(\mathbb{Z}_4\)
zbMATH Open document ID: 1054.94015
DOI: 10.1023/A:1027338603957
author: P. Vijay Kumar
author: Dong-Joon Shin
author: Tor Helleseth
published in: Designs, Codes and Cryptography
publication date: 2004-06-11T00:00:00Z
review text: From the supports of codewords of a code, which have a same support size \(\tau\), it is sometimes possible to construct a \(t\)-\((n,\tau,\lambda)\)-design. The Assmus-Mattson theorem of \textit{E. F. Assmus, Jr.} and \textit{H. F. Mattson, Jr.} [J. Comb. Theory 6, 122--151 (1969; Zbl 0179.02901)] gives a sufficient condition for the codewords of minimum Hamming weight in a code over a finite field to form a simple \(t\)-design. The authors determine in this article the complete weight enumerator of the Delsarte-Goethals code over \(\mathbb{Z}_4\). They also present an Assmus-Mattson type approach to identify \(t\)-designs in linear codes over \(\mathbb{Z}_4\). This approach, in combination with the complete weight enumerator, is used to show that the codewords of constant Hamming weight in the Goethals code over \(\mathbb{Z}_4\) as well as in the Delsarte-Goethals code over \(\mathbb{Z}_4\) yield 3-designs, possibly with repeated blocks. More precisely, for the Goethals code of length \(2^m\), with odd \(m \geq 5\), the supports of codewords of the same Hamming weight greater than or equal to 8 form a 3-design, possibly with repeated blocks. Similarly, for the Goethals-Delsarte code of length \(2^m\), with odd \(m\geq 5\), the supports of codewords of the same Hamming weight smaller than or equal to \(2^m-3\) form a 3-design, possibly with repeated blocks. A special case of their Assmus-Mattson approach, used for the Hamming weight enumerator, enables them to derive an Assmus-Mattson type theorem for linear codes over \(\mathbb{Z}_4\).
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 2073423
zbMATH Keywords: t-designs
zbMATH Keywords: \(Z_4\) codes
zbMATH Keywords: Assmus-Mattson theorem
zbMATH Keywords: complete weight enumerator
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: a:1027338603957
OpenAlex ID: W1497223756
description: scientific article; zbMATH DE number 2073423