On a pair of dual subschemes of the Hamming scheme \(H_ n(q)\)
An association scheme with n classes an a set X is a partition of the set of 2-element subsets of X into n classes \(\Gamma_ 1,\Gamma_ 2,...,\Gamma_ n\) satisfying (1) given \(x\in X\), the number \(v_ i\) of \(y\in X\) with \(\{x,y\}\in \Gamma_ i\) depends only on i; (2) given x,y\(\in X\) with \(\{x,y\}\in \Gamma_ k\), the number of \(z\in X\) with \(\{x,z\}\in \Gamma_ i\) and \(\{y,z\}\in \Gamma_ j\) is a constant \(P^ k_{i,j}\) depending only on i,j,k. The Hamming scheme \(H_ n(q)\) is an association scheme with n classes, where \(X=F^ n\), \(F=GF(q)\) and a pair of vectors \(\{\) x,y\(\}\) is in \(\Gamma_ i\) if and only if the Hamming distance \(d(x,y)=i.\) The authors analyze codes in the Hamming association scheme \(H_ n(q)\). The main result is Theorem 1: Let C be a subset of \(H_ n(q)\) with degree \(e+1\) and strength 2e or \(2e+1\). If \(w_ 1,w_ 2,...,w_{e+1}\) are nonzero distances between elements of C then \[ q^{e(e+1)/2}\prod_{i>j}(w_ i-w_ j)/\prod^{e}_{k=1}(k!) \] is an integer dividing \(| C|^{e+1}\). Further, the authors give necessary conditions for the existence of nonlinear uniformly packed codes, and they determine the possible parameters of uniformly packed 2- error-correcting codes.
scientific article; zbMATH DE number 3926939
scientific article