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ACD codes over skew-symmetric dualities

Astha Agrawal, R. K. Sharma

TL;DR

We address the problem of characterizing additive complementary dual (ACD) codes under arbitrary dualities and introduce a skew-symmetric subclass. The authors establish a criterion that an additive code $C$ is ACD with respect to a duality $M$ iff the matrix $\log_{\xi}(\mathcal{G} \odot_M \mathcal{G}^T)$ is invertible over $\mathbb{F}_p$, prove closure properties of dualities, count symmetric versus skew-symmetric dualities, and derive lower bounds on the minimum distance for ACD codes on skew-symmetric dualities. They also show that the generator count must be even for ACD codes in the skew-symmetric class and present constructions yielding new quaternary ACD codes with improved parameters, including non-symmetric dualities over $\mathbb{F}_4$. The results extend ACD code design to broad duality classes and indicate practical gains for quantum coding applications.

Abstract

The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the implementation of a variety of dualities. The article begins by developing the properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we introduce a subclass of non-symmetric dualities referred to as skew-symmetric dualities. Then, we precisely count symmetric and skew-symmetric dualities over finite fields. Two conditions have been obtained: one is a necessary and sufficient condition, and the other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities. The necessary condition is on the generator matrix of an ACD code over skew-symmetric dualities. We provide bounds for the highest possible minimum distance of ACD codes over skew-symmetric dualities. Finally, we find some new quaternary ACD codes over non-symmetric dualities with better parameters than the symmetric ones.

ACD codes over skew-symmetric dualities

TL;DR

We address the problem of characterizing additive complementary dual (ACD) codes under arbitrary dualities and introduce a skew-symmetric subclass. The authors establish a criterion that an additive code is ACD with respect to a duality iff the matrix is invertible over , prove closure properties of dualities, count symmetric versus skew-symmetric dualities, and derive lower bounds on the minimum distance for ACD codes on skew-symmetric dualities. They also show that the generator count must be even for ACD codes in the skew-symmetric class and present constructions yielding new quaternary ACD codes with improved parameters, including non-symmetric dualities over . The results extend ACD code design to broad duality classes and indicate practical gains for quantum coding applications.

Abstract

The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the implementation of a variety of dualities. The article begins by developing the properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we introduce a subclass of non-symmetric dualities referred to as skew-symmetric dualities. Then, we precisely count symmetric and skew-symmetric dualities over finite fields. Two conditions have been obtained: one is a necessary and sufficient condition, and the other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities. The necessary condition is on the generator matrix of an ACD code over skew-symmetric dualities. We provide bounds for the highest possible minimum distance of ACD codes over skew-symmetric dualities. Finally, we find some new quaternary ACD codes over non-symmetric dualities with better parameters than the symmetric ones.
Paper Structure (8 sections, 31 theorems, 9 equations, 2 tables)

This paper contains 8 sections, 31 theorems, 9 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ACD codes over arbitrary dualities
  4. Skew-symmetric dualities
  5. ACD codes over skew-symmetric dualities
  6. Bounds of ACD codes on skew-symmetric dualities
  7. Non-symmetric dualities over $\mathbb{F}_4$
  8. Conclusion

Key Result

Lemma 1

ACDgroupcharacter Let $C$ and $D$ be ACD codes of lengths $n$ and $m$, respectively, over a group $G$ with respect to duality $M$. Then $C\times D$ is an ACD code with respect to duality $M$ of length $n + m$.

Theorems & Definitions (77)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Definition 5
  • Definition 6
  • Lemma 1
  • Theorem 2
  • Remark
  • ...and 67 more