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F. I. Solov'eva

Publications and source records attributed to F. I. Solov'eva.

11 recordsLinked to original sources
arXiv2020

Coordinate transitivity of extended perfect codes and their SQS

We continue the study of the class of binary extended perfect propelinear codes constructed in the previous paper and consider their permutation automorphism (symmetry) groups and Steiner quadruple systems. We show that the automorphism group of the SQS of any such code coincides with the permutation automorphism group of the code. In particular, the SQS of these codes are complete invariants for the isomorphism classes of these codes. We obtain a criterion for the point transitivity of the automorphism group of SQS of proposed codes in terms of GL-equivalence (similar to EA-type equivalence for permutations of F^r). Based on these results we suggest a new construction for coordinate transitive and neighbor transitive extended perfect codes.

cs.IT↗
arXiv2019

On existence of perfect bitrades in Hamming graphs

A pair $(T_0,T_1)$ of disjoint sets of vertices of a graph $G$ is called a perfect bitrade in $G$ if any ball of radius 1 in $G$ contains exactly one vertex in $T_0$ and $T_1$ or none simultaneously. The volume of a perfect bitrade $(T_0,T_1)$ is the size of $T_0$. In particular, if $C_0$ and $C_1$ are distinct perfect codes with minimum distance $3$ in $G$ then $(C_0\setminus C_1,C_1\setminus C_0)$ is a perfect bitrade. For any $q\geq 3$, $r\geq 1$ we construct perfect bitrades in the Hamming graph $H(qr+1,q)$ of volume $(q!)^r$ and show that for $r=1$ their volume is minimum.

cs.IT↗
arXiv2019

A concatenation construction for propelinear perfect codes from regular subgroups of GA(r,2)

A code $C$ is called propelinear if there is a subgroup of $Aut(C)$ of order $|C|$ acting transitively on the codewords of $C$. In the paper new propelinear perfect binary codes of any admissible length more than $7$ are obtained by a particular case of the Solov'eva concatenation construction--1981 and the regular subgroups of the general affine group of the vector space over $GF(2)$.

math.CO↗
arXiv2018

On components of a Kerdock code and the dual of the BCH code $C_{1,3}$

In the paper we investigate the structure of $i$-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length $n=2^m-1$, for odd $m$. We prove that for any admissible length a punctured Kerdock code consists of two $i$-components and the dual of BCH code is a $i$-component for any $i$. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restriction of the Hamming scheme to a doubly shortened Kerdock code is an association scheme.

math.CO↗
arXiv2014

On homogeneous nontransitive binary perfect code

Studying binary perfect codes we show the existence of homogeneous nontransitive codes. Thus, as far as perfect codes are concerned, the propelinear codes are strictly contained in transitive codes, wheresas homogeneous codes form a strict subclass of transitive codes. In the work we deduce a necessary and sufficient condition for transitivity of perfect binary codes of rank one more than that of Hamming code. The paper is in Russian.

math.CO↗
arXiv2014

Transitive nonpropelinear perfect codes

A code is called transitive if its automorphism group (the isometry group) of the code acts transitively on its codewords. If there is a subgroup of the automorphism group acting regularly on the code, the code is called propelinear. Using Magma software package we establish that among 201 equivalence classes of transitive perfect codes of length 15 from \cite{ost} there is a unique nonpropelinear code. We solve the existence problem for transitive nonpropelinear perfect codes for any admissible length $n$, $n\geq 15$. Moreover we prove that there are at least 5 pairwise nonequivalent such codes for any admissible length $n$, $n\geq 255$.

math.CO↗
arXiv2013

On the number of nonequivalent propelinear extended perfect codes

The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation.

math.CO↗
arXiv2012

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n=2^m-1 for small m, m < 23, is given. As far as we know, for m in {5,7,11,13,17,19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m < 21. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven.

cs.IT↗
arXiv2008

Construction of Z4-linear 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 and fundamental properties as the codes in the usual binary linear Reed-Muller family. To make more evident the duality relationships in the constructed families the concept of Kronecker inner product is introduced.

cs.IT↗