Faina I. Solov'eva

Structural properties of binary propelinear codes

The paper deals with some structural properties of propelinear binary codes, in particular propelinear perfect binary codes. We consider the connection of transitive codes with propelinear codes and show that there exists a binary code, the Best code of length 10, size 40 and minimum distance 4, which is transitive but not propelinear. We propose several constructions of propelinear codes and introduce a new large class of propelinear perfect binary codes, called normalized propelinear perfect codes. Finally, based on the different values for the rank and the dimension of the kernel, we give a lower bound on the number of nonequivalent propelinear perfect binary codes.

On the Ranks and Kernels Problem for Perfect Codes

A construction is proposed which, for n large enough, allows one to build perfect binary codes of length n and rank r, with kernel of dimension k, for any admissible pair (r, k) within the limits of known bounds.

On the Structure of Symmetry Groups of Vasil'ev Codes

The structure of symmetry groups of Vasil’ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil’ev code of length n is always nontrivial; for codes of rank n − log(n + 1) +1, an attainable upper bound on the order of the symmetry group is obtained.

Structure of i-components of perfect binary codes☆

Special components of perfect binary codes are investigated. We call such components i-components. A class of perfect codes of length n with minimal i-components of cardinality (k+1)2n−k/(n+1) for every n=2s−1,s>2 and k=2r−1, where r=2,…,s−1 is constructed. The existence of maximal cardinality nonisomorphic i-components of different perfect codes of length n for all n=2s−1,s>3, is proved.

On ℤ4-linear codes with the parameters of Reed-Muller codes

For any pair of integers r and m, 0 ≤ r ≤ m, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical rth-order Reed-Muller code RM(r, m).

On perfect binary codes

Some results on perfect codes obtained during the last 6 years are discussed. The main methods to construct perfect codes such as the method of @a-components and the concatenation approach and their implementations to solve some important problems are analyzed. The solution of the ranks and kernels problem, the lower and upper bounds of the automorphism group order of a perfect code, spectral properties, diameter perfect codes, isometries of perfect codes and codes close to them by close-packed properties are considered.

Intersection matrices for partitions by binary perfect codes

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?.

To the Metrical Rigidity of Binary Codes

A code C in the n-dimensional metric space En over GF(2) is called metrically rigid if each isometry I : C → En can be extended to an isometry of the whole space En. For n large enough, metrical rigidity of any length-n binary code that contains a 2-(n, k, λ)-design is proved. The class of such codes includes, for instance, all families of uniformly packed codes of large enough lengths that satisfy the condition d − ρ ≥ 2, where d is the code distance and ρ is the covering radius.

Quaternary Plotkin constructions and quaternary Reed-Muller codes

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 codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters as the codes in the classical binary linear Reed-Muller family.

Reconstructing Extended Perfect Binary One-Error-Correcting Codes From Their Minimum Distance Graphs

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.