Association schemes related to Kasami codes and Kerdock sets
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.
scientific article; zbMATH DE number 1548361
scientific article