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We examine designs and binary codes associated with the line graph of the n-cube Qn, i.e. the Hamming graph H(n, 2). We find the automorphism groups and the parameters of the codes. We find a regular subgroup of the automorphism group that can be used for permutation decoding, or partial permutation decoding, for any information set.

We show how to find s-PD-sets of size s + 1 that satisfy the Gordon-Schönheim bound for partial permutation decoding for the binary simplex codes S n (F 2) for all n ≥ 4, and for all values of s up to 2 n −1 n − 1. The construction also applies to the q-ary simplex codes S n (F q) for q > 2, and to s-antiblocking information systems of size s + 1, for s up… (More)

For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γ k n has vertices the 2 n vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γ 1 n is the n-cube, usually denoted by Q n. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following… (More)

We examine the binary codes C2(Ai + I) from matrices Ai + I where Ai is an adjacency matrix of a uniform subset graph Γ(n, 3, i) of 3-subsets of a set of size n with adjacency defined by subsets meeting in i elements of Ω, where 0 ≤ i ≤ 2. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the C2(Ai + I) are also examined.… (More)

Let $$n, m \ge 2$$ n , m ≥ 2 be integers. The cartesian, categorical and lexicographic products of m copies of the n-cycle denoted by $$C_n$$ C n all have as their vertex-set $$\{0, 1, \ldots , n-1\}^m$$ { 0 , 1 , … , n - 1 } m , with adjacency defined variously. In this paper the binary codes generated by the row span of adjacency matrices of the… (More)

The Johnson graph, denoted by J(n, k), is the graph of which the vertex-set is the set of all k-subsets of Ω = {1, 2,. .. , n}, and any two vertices u and v constitute an edge [u, v] if and only if |u ∩ v| = k − 1. In this talk the codes and their duals generated by the adjacency matrix of J(n, k) will be described. It will be shown that in each case, the… (More)

We examine the ternary codes $$C_3(A_i+I)$$ C 3 ( A i + I ) from matrices $$A_i+I$$ A i + I where $$A_i$$ A i is an adjacency matrix of a uniform subset graph $$\Gamma (n,3,i)$$ Γ ( n , 3 , i ) of $$3$$ 3 -subsets of a set of size $$n$$ n with adjacency defined by subsets meeting in $$i$$ i elements of $$\Omega $$ Ω , where $$0 \le i \le 2$$ 0 ≤ i ≤ 2 .… (More)