A family of constacyclic codes over a class of non-chain rings A q,r and new quantum codes

Habibul Islam (1) , Shikha Patel (1) , Om Prakash (1) , Patrick Solé (2)

1 IITP - Indian Institute of Technology Patna
2 I2M - Institut de Mathématiques de Marseille

For a prime p, let F q 2 be the finite field of q 2 elements where q = p m and m ≥ 1 is an integer. In this paper, we study constacyclic and skew constacyclic codes of length n over a class of finite commutative non-chain rings A q,r = F q 2 [u 1 , u 2 ,. .. , u r ]/ u 2 i − γ i u i , u i u j = u j u i = 0 where 1 ≤ i = j ≤ r, γ i ∈ F * q 2 , and r ≥ 1 is an integer. For a unit Λ = r i=0 κ i Λ i in the ring A q,r , we show that a (skew) Λ-constacyclic code of length n is a direct sum of (skew) Λ i-constacyclic codes of length n over F q 2. Also, we derive the necessary and sufficient conditions for such codes (constacyclic and skew constacyclic) to contain their Hermitian duals. By applying the Hermitian construction on the Gray images of dual containing codes, many MDS and new quantum codes with parameters better than the best-known codes are constructed. Keywords Non-chain ring • Constacyclic code • Skew constacyclic code • Gray map • Hermitian construction • Quantum code.

Mots clés: Non-chain ring; Constacyclic code; Skew constacyclic code; Gray map; · Hermitian construction; Quantum code.

Domaines: Théorie de l'information [cs.IT]

Habibul Islam, Shikha Patel, Om Prakash, Patrick Solé. A family of constacyclic codes over a class of non-chain rings A q,r and new quantum codes. Journal of Applied Mathematics and Computing, 2021. ⟨hal-03324685⟩