Arithmetic completely regular codes

Jacobus H. Koolen, Woo Sun Lee, William J. Martin, Hajime Tanaka

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.

Original languageEnglish
Pages (from-to)59-76
Number of pages18
JournalDiscrete Mathematics and Theoretical Computer Science
Volume17
Issue number3
Publication statusPublished - 2016 Jan 1

Keywords

  • Completely regular code
  • Coset graph
  • Distance-regular graph
  • Hamming graph
  • Leonard's Theorem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Discrete Mathematics and Combinatorics

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