The Quasi-self-dual, Right LCD and ACD codes over a noncommutative non-unital ring

Quasi-self-dual, Right LCD and ACD codes over a noncommutative non-unital ring

  • Anup Kushwaha Indian Institute of Technology Patna
  • Shikha Yadav Indian Institute of Technology Patna
  • Om Prakash
Keywords: Additive codes; Linear codes; QSD codes; LCD codes; ACD codes

Abstract

This paper presents the study of QSD (quasi-self-dual), right-LCD (linear complementary dual), and ACD (additive complementary dual) codes over a noncommutative local ring $R= \langle a,b ~|~ 3a =3b=0,~ a^2=a,~ b^2=b,~ ab=b,~ ba=a \rangle$ of order $9$. Initially, over this ring $R$, we introduce QSD codes and characterize their multilevel construction. Then, we delve into the study of right LCD codes over the ring $R$ and demonstrate a method for constructing these codes based on ternary LCD codes. Finally, we introduce the right-ACD codes over this ring and present several criteria for the existence of such codes.

Author Biographies

Anup Kushwaha, Indian Institute of Technology Patna

Department of Mathematics

Indian Institute of Technology Patna

Shikha Yadav, Indian Institute of Technology Patna

Department of Mathematics

Indian Institute of Technology Patna

Published
2024-09-05
Section
Articles