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Some $3$-designs invariant under $2.PΣL(2,49).$

Minjia Shi, Ruowen Liu, Patrick Solé

TL;DR

The paper constructs and analyzes ternary abelian codes derived from Jacobsthal matrices to produce extended Generalized Quadratic Residue codes with automorphism groups related to $PΣL(2,49)$. It demonstrates that the union of a code and its dual yields 3-designs for several weights, and provides a group-action explanation for these designs beyond standard transitivity arguments. The study extends these ideas to the extended GQR code of length $26$ and to extended QR codes of lengths $14$ and $38$, showing systematic emergence of 3-designs across multiple weights. The results highlight a ternary analogue of existing binary phenomena and open questions about generalizing the orbit-based explanations to arbitrary $q$.

Abstract

We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and its dual supports a 3-(50,14,1248) design. The automorphism group of the latter design is a double cover of the permutation part of the automorphism group of the [50,25,8] code, which is isomorphic to $PΣL(2,49).$ Other weights in this code, other GQR codes, and other QR codes yield other 3-designs by the same process. A simple group action argument is provided to explain this behaviour of isodual codes.

Some $3$-designs invariant under $2.PΣL(2,49).$

TL;DR

The paper constructs and analyzes ternary abelian codes derived from Jacobsthal matrices to produce extended Generalized Quadratic Residue codes with automorphism groups related to . It demonstrates that the union of a code and its dual yields 3-designs for several weights, and provides a group-action explanation for these designs beyond standard transitivity arguments. The study extends these ideas to the extended GQR code of length and to extended QR codes of lengths and , showing systematic emergence of 3-designs across multiple weights. The results highlight a ternary analogue of existing binary phenomena and open questions about generalizing the orbit-based explanations to arbitrary .

Abstract

We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and its dual supports a 3-(50,14,1248) design. The automorphism group of the latter design is a double cover of the permutation part of the automorphism group of the [50,25,8] code, which is isomorphic to Other weights in this code, other GQR codes, and other QR codes yield other 3-designs by the same process. A simple group action argument is provided to explain this behaviour of isodual codes.
Paper Structure (14 sections, 7 theorems, 20 equations, 6 tables)

This paper contains 14 sections, 7 theorems, 20 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Ternary codes
  4. Designs
  5. Permutation groups
  6. Isodual codes
  7. A family of ternary abelian Codes
  8. The extended GQR code of length $50$
  9. Other codes
  10. The extended GQR code of length $26$
  11. The extended QR codes of length $14$ and $38$
  12. The extended QR code of length 14
  13. The extended QR code of length 38
  14. Conclusion

Key Result

Lemma 2.1

The group $Per(C)$ permutes the supports of the codewords of $C.$ The dual$C^\perp$ of a code $C$ is understood with respect to the standard inner product $(.).$ A ternary code is isodual if it is equivalent to its dual, and self-dual if it is equal to its dual. A cyclic ternary code of length $n$ is an ideal in $\frac{\mathbb{F}_3[x]}{(x^n-1)}$ of the form $(g(x)).$ When $n=p,$ an odd prime,

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • proof
  • ...and 2 more