Association schemes related to Kasami codes and Kerdock sets
type: Item
instance of: scholarly article
title: Association schemes related to Kasami codes and Kerdock sets
zbMATH Open document ID: 0954.05052
DOI: 10.1023/A:1008385102731
published in: Designs, Codes and Cryptography
publication date: 2001-01-01T00:00:00Z
review text: For a positive number \(t\), the Kasami code \(K_t\) consists of all subsets \(S\) of \(\text{GF} (2^{2t+1})-\{0\}\) such that \(\sum_{r\in S}r=\sum_{r\in S}r^3=0\). The vertices of the Kasami graph \(\Gamma(K_t)\) are all ordered pairs of elements in \(\text{GF} (2^{2t+1})\) and two distinct pairs \((a,x)\) and \((b,y)\) are adjacent if and only if \(a+b=(x+y)^3\). Theorem 2. Define five relations on the set of all ordered pairs of elements of \(\text{GF} (2^{2t+1})\) as follows. For distinct pairs \((a,x)\) and \((b,y)\), the possible relations are \(G_1\): \(x\neq y\) and \(a+b=(x+y)^3\); \(G_2\): \(x\neq y\) and \(a+b=xy(x+y)\); \(G_3\): \(x\neq y\) and \(a+b\neq (x+y)^3\) and \(\text{Tr} ((a+b)/(x+y)^3)=1\); \(G_4\): \(x\neq y\) and \(a+b\neq xy(x+y)\) and \(\text{Tr} ((a+b)/(x+y)^3)=0\); \(G_5\): \(x=y\) and \(a\neq b\). Then the relations \(G_i\), together with the identity relation \(G_0\), form an association scheme. The formally dual scheme for this association scheme is constructed in Theorem 8.
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 11H71
zbMATH DE Number: 1548361
zbMATH Keywords: association scheme
zbMATH Keywords: binary code
zbMATH Keywords: quadric
author: D. de Caen
author: Edwin R. Van Dam
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
Wikidata QID: Q56850761
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description: scientific article; zbMATH DE number 1548361