New quantum codes from constacyclic and additive constacyclic codes
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New quantum codes from constacyclic and additive constacyclic codes Habibul Islam1 · Om Prakash1 Received: 10 April 2020 / Accepted: 17 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let p be a prime and q = pr , for an integer r ≥ 1. This article studies λ = (λ1 + uλ2 +vλ3 )-constacyclic codes of length n over a class of finite commutative non-chain rings R = Fq [u, v]/u 2 − γ u, v 2 − δv, uv = vu = 0, where γ , δ ∈ Fq∗ . First, we decompose (λ1 +uλ2 +vλ3 )-constacyclic code into the direct sum of λ1 -constacyclic, (λ1 +γ λ2 )-constacyclic and (λ1 +δλ3 )-constacyclic codes over Fq , respectively. Then, we determine the necessary and sufficient condition for these codes to contain their Euclidean duals. Further, we extend the study to Fq R-additive λ-constacyclic codes of length (n, m) which are R[x]-submodules of Sn,m = Fq [x]/x n −1× R[x]/x m −λ. Apart from other results, we also discuss the dual-containing separable Fq R-additive λ-constacyclic codes. Finally, by using the CSS construction on the Gray images of these codes, we obtain many new and better quantum codes that improve on the known existing quantum codes available in recent articles. Keywords Non-chain ring · Constacyclic code · Gray map · Additive code · Quantum code Mathematics Subject Classification 94B05 · 94B15 · 94B35 · 94B60
1 Introduction Like classical linear codes, quantum codes help to protect quantum information during transmission through a quantum channel. These codes have been extensively used in quantum computation which solves challenging problems faster than the classical computation. For instance, the running time of the Shor’s Algorithm [36] to find the
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Om Prakash [email protected] Habibul Islam [email protected]
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Department of Mathematics, Indian Institute of Technology Patna, Patna 801 106, India 0123456789().: V,-vol
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prime factors of a large integer is polynomial time in quantum computation, whereas sub-exponential in classical computation. Quantum code was introduced by Shor [35]. Later, Calderbank et al. [9] constructed the quantum codes from classical codes in a formal way. Recall that a q-ary quantum code denoted by [[n, k, d]]q satisfies the quantum singleton bound 2d + k ≤ n + 2 and called maximum distance separable (MDS) if the bound is attained. Clearly, MDS codes have the best error control (highest distance) and best code rate (larger non-redundant bits) compared to the other codes with the same parameters. But, these codes are rare to find. Hence, people have been constructing quantum codes close to MDS and store them to some online platform, like [16]. Usually, to validate the novelty of the approach, researchers are comparing their obtained codes to the codes which are known by most recent articles. Kai and Zhu [26] determined the quantum codes over F4 from the cyclic codes over F4 + uF4 . Qian [31] obtained binary quantum codes by using cyclic codes over F2 + vF2 . Later, the study of cyclic codes over finit
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