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On linear lexicographic codes: Ninth column construction of the ternary Golay code

Yuki Irie

Abstract

We characterize linear lexicographic $p$-ary codes. Using this characterization, when $p \ge 3$, we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain minimum distances. In these excluded cases, we may obtain linear codes of higher dimensions; for instance, when $p = 3$ and $d = 6$, the ternary Golay code is obtained.

On linear lexicographic codes: Ninth column construction of the ternary Golay code

Abstract

We characterize linear lexicographic -ary codes. Using this characterization, when , we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain minimum distances. In these excluded cases, we may obtain linear codes of higher dimensions; for instance, when and , the ternary Golay code is obtained.
Paper Structure (22 sections, 52 theorems, 268 equations)

This paper contains 22 sections, 52 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Lexicographic codes
  3. Linearity of lexicographic codes
  4. Dimensions of linear lexicographic codes
  5. Organization
  6. A characterization of linear lexicographic codes
  7. Griesmer bound
  8. Lexicographic codes meeting the Griesmer bound
  9. Ternary linear lexicographic codes
  10. Stronger versions of lemmas in Section 4
  11. Linearity breaking
  12. Ternary linear lexicographic codes
  13. Solomon-Stiffler codes
  14. pi-d-distributed matrices
  15. Restriction
  16. ...and 7 more sections

Key Result

Theorem 1.2

If ${p} = 2$, then ${\mathcal{X}}^{; {k}}$ is a linear code.

Theorems & Definitions (110)

  • Example 1.1
  • Theorem 1.2: conway-lexicographic-1986levenshtein-class-1960
  • Example 1.3: conway-lexicographic-1986
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 1.8: conway-atlas-1986
  • Example 1.9
  • Theorem 1.10
  • ...and 100 more