It is known [4] that for any numbers n = 2m- 1, m ≥ 4 and r, such that n - log(n + 1) ≤ r ≤ n there exists a perfect binary code of length n and rank r. We show that there exists a propelinear such code of length n, excluding, may be, n = r = 63, n = 127, r ϵ {126,127} and n = r = 2047.

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2m- 1 for small m (m ≤ 22), is given. For m ∈{5, 7,11,13,17,19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is proposed. This classification for APN monomial power permutations coincides with the CCZ-equivalence, at least up to m ≤ 17.

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.

In this paper, new quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension, and the minimum distance, are studied. Using these constructions, new families of quaternary Reed-Muller (RM) codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. To make the duality relationships in the constructed families more evident, the concept of Kronecker inner product is introduced.

The intersection structure forZ2Z4-additive Hadamard codes is investigated. For any two of these codesC1andC2, the Abelian group structure of the intersectionC1capC2is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed, establishing lower and upper bounds for them. Constructions are given of codes whose intersection has any parameters between these bounds.

The intersection problem for Z2Z4-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z2Z4-additive codes C1and C2of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these Z2Z4-additive codes C1and C2, the abelian group structure of the intersection codes C1cap C2is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given.

We investigate the following problem: given two partitions of the Hamming space, their intersection matrix provides the cardinalities of the pairwise intersections of the subsets of these partitions. If we consider partitions by extended perfect codes, how many intersection matrices can we construct?.