On homogeneous nontransitive binary perfect code
I.Yu.Mogilnykh, F.I. Solov'eva
TL;DR
This paper investigates the structure of binary perfect codes, revealing the existence of homogeneous nontransitive codes and establishing conditions for transitivity in certain perfect codes.
Contribution
It introduces the concept of homogeneous nontransitive codes and provides a necessary and sufficient condition for the transitivity of specific perfect binary codes.
Findings
Homogeneous nontransitive codes exist among binary perfect codes
Propelinear codes are a strict subset of transitive codes
A characterization of transitivity for perfect codes of rank one more than Hamming code
Abstract
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.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Cooperative Communication and Network Coding