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Quantum codes from a class of constacyclic codes over finite commutative rings

Authors: 

Hai Q. Dinh, Tushar Bag, Ashish K. Upadhyay, Mohammad Ashraf, Ghulam Mohammad, Warattaya Chinnakum

Source title: 
Journal of Algebra and Its Applications, 19(12): 2150003, 2020 (ISI)
Academic year of acceptance: 
2020-2021
Abstract: 

Let p be an odd prime, and kk be an integer such that gcd(k, p) = 1. Using pairwise orthogonal idempotents γ1, γ2, γ3 of the ring = 𝔽p[u]/〈uk+1u〉, with γ+ γ+ γ= 1, ℛ is decomposed as ℛ = γ1γ2γ3, which contains the ring R = γ1𝔽pγ2𝔽pγ3𝔽p as a subring. It is shown that, for λ0, λ∈ 𝔽pλ0+ukλR, and it is invertible if and only if λ0 and λ0 + λkare units of 𝔽p. In such cases, we study (λ0+ukλk)-constacyclic codes over RR. We present a direct sum decomposition of (λ+ ukλk)-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a (λ+ ukλk)-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over 𝔽p, with better parameters than existing ones, are constructed from cyclic and negacyclic codes over R.