Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The punctured dodecacode is unique

Markus Grassl, Denis Krotov, Lin Sok, Patrick Solé

Abstract

The punctured dodecacode is an additive $4$-ary code of length $11$ and distance $5$ which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also establish the nonexistence of analogues of the dodecacode and the punctured dodecacode in Doob graphs. To that end, we classify two-weight codes of weights $6$ and $8$ in Doob and $4$-ary Hamming graphs of diameter $9$ and the corresponding strongly regular graphs. Keywords: dodecacode, additive code, trace Hermitian duality, uniformly packed code, completely regular code, Doob graph, strongly regular graph

The punctured dodecacode is unique

Abstract

The punctured dodecacode is an additive -ary code of length and distance which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also establish the nonexistence of analogues of the dodecacode and the punctured dodecacode in Doob graphs. To that end, we classify two-weight codes of weights and in Doob and -ary Hamming graphs of diameter and the corresponding strongly regular graphs. Keywords: dodecacode, additive code, trace Hermitian duality, uniformly packed code, completely regular code, Doob graph, strongly regular graph
Paper Structure (33 sections, 4 theorems, 38 equations, 7 tables)

This paper contains 33 sections, 4 theorems, 38 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Codes in Hamming metric space
  3. Preliminaries
  4. Results
  5. Partial classification results
  6. Codes in Doob metric space
  7. Preliminaries
  8. Two-weight codes of size 64
  9. Lengthening to codes in Doob graphs of diameter 11 and 12
  10. Conclusion and open problems
  11. All weight-{6,8} codes in D(m,n) of diameter 9
  12. In D(4,1+0)
  13. In D(3,3+0)
  14. In D(4,0+1)
  15. In D(3,2+1)
  16. ...and 18 more sections

Key Result

Proposition 3.1

In $D(m,n'+n")$, $2m+n'+n"=9$, there are exactly $26$ equivalence classes of weight-$\{6,8\}$ codes of size $64$, $B_1$, …, $B_6$ in $D(4,1+0)$ (Table t:410), $C_1$, …, $C_8$ in $D(3,0+3)$ (Table t:303), $D_1$, …, $D_8$ in $D(0,9+0)=H(9,4)$ (Table t:090). The weight distribution of all these codes i

Theorems & Definitions (10)

  • proof
  • proof
  • proof
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Corollary 3.4
  • proof
  • Proposition 3.5
  • proof