If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of $Z_4$-linear Hadamard codes is given.Comment: 7p. WCC-200

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k − 1, k, v) bitrades, extended 1-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of q-ary Steiner T q (k − 1, k, v) bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph J q (v, k).

A new generalization of the Gray map is introduced. The new generalization $\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized Gray map $\phi$ in the following way: if we take two dual linear $Z_{2^k}$-codes and construct binary codes from them using the generalizations $\phi$ and $\Phi$ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of $Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect codes are described, where co-$Z_{2^k}$-linearity means that the code can be obtained from a linear $Z_{2^k}$-code with the help of the new generalized Gray map. Keywords: Gray map, Hadamard codes, MacWilliams identity, perfect codes, $Z_{2^k}$-linearityComment: English: 10pp, Russian: 14pp; V.1 title: Z_{2^k}-duality, Z_{2^k}-linear Hadamard codes, and co-Z_{2^k}-linear 1-perfect codes; V.2: revised; V.3: minor revision, references updated, Russian translation adde

A q-ary code of length n, size M , and minimum distance d is called an (n, M, d)q code. An (n, q k , n − k + 1)q code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every7} is equivalent to a linear code with the same parameters. This implies that the (6, 5 4 , 3)5 code and the (n, 7 n−2 , 3)7 MDS codes for n ∈ {6, 7, 8} are unique. The classification of one-errorcorrecting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8 n−2 , 3)8 codes for n = 6, 7, 8, 9, respectively. One of the equivalence classes of perfect (9,8 7 , 3)8 codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.