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New Quantum codes from skew constacyclic codes Over a Class Of Non-Chain Rings R e, q

INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS 2021年第9期60卷 3334-3352页

作者： Prakash, Om Islam, Habibul Patel, Shikha Sole, Patrick Indian Inst Technol Patna Dept Math Patna 801106 Bihar India Aix Marseille Univ Cent Marseille CNRS I2M Marseille France

In this paper, for q = p(m) (p is prime) such that q equivalent to 1 (mod e), we study skew constacyclic codes over a class of non-chain rings R-e,R-q = F-q[u]/ where m, e >= 2 are integers. We decompose the ring i... 详细信息

In this paper, for q = p(m) (p is prime) such that q equivalent to 1 (mod e), we study skew constacyclic codes over a class of non-chain rings R-e,R-q = F-q[u]/< u(e) - 1 > where m, e >= 2 are integers. We decompose the ring into a direct sum of local rings, and consequently, skew constacyclic codes over that ring into a direct sum of skew constacyclic codes over local rings. This decomposition yields the structure of Euclidean duals of skew constacyclic codes and further a necessary and sufficient condition to contain their duals. From an application point of view, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing skew constacyclic codes and obtain many quantum codes improving the best-known codes in the literature.

关键词： skew polynomial ring skew constacyclic code Gray map CSS construction Quantum code

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A note on skew constacyclic codes over F_q

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2019年第3期11卷

作者： Islam, Habibul Prakash, Om Indian Inst Technol Patna Dept Math Patna 801106 Bihar India

In this paper, the skew constacyclic codes over finite non-chain ring R = F-q + uF(q) + vF(q), where q = p(m), p is an odd prime and u(2) = u, v(2) = v, uv = vu = 0, are studied. We show that the Gray image of a skew alpha-constacyclic, code of length n over R. is a skew quasi-twisted code of length 3n over F-q of index 3. Further, the structural properties of skew constacyclic codes over R are obtained by the decomposition method.

关键词： skew polynomial ring skew constacyclic code quasi-twisted code Gray map idempotent generator non-chain ring

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Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring

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QUANTUM INFORMATION PROCESSING 2023年第5期22卷 200-200页

作者： Prakash, Om Verma, Ram Krishna Singh, Ashutosh Indian Inst Technol Patna Dept Math Patna 801106 India SRM Inst Sci & Technol Delhi NCR Campus Ghaziabad 201204 Uttar Pradesh India

For a prime p and a positive integer e, let Fpe be the finite field and R := Fpe [u, v]/ f (u), g(v), uv - vu, where f (u) and g(v) are non-constant square-free polynomials of degree r and s, respectively. This paper constructs quantum and linear complementary dual (briefly, LCD) codes from skew constacyclic codes over the ring R. Toward this, we first discuss the explicit structure of skew constacyclic codes and their Euclidean as well as Hermitian duals over R. Then, we establish a necessary and sufficient condition for these codes to contain their Euclidean (Hermitian) duals. Further, by applying CSS (Hermitian) construction, many new quantum codes with better parameters are obtained. Moreover, a necessary and sufficient condition is established for these codes over R to be Euclidean (Hermitian) LCD. Finally, many examples of MDS codes over Fpe are provided under the gray images of the skew Euclidean LCD codes.

关键词： constacyclic code skew constacyclic code Additive code Quantum code Gray map

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A class of constacyclic codes over Z4[u]/⟨u^k⟩

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2019年第1-2期60卷 237-251页

作者： Islam, Habibul Bag, Tushar Prakash, Om Indian Inst Technol Patna Dept Math Patna 801106 Bihar India

In this paper, we study -constacyclic and skew -constacyclic codes over the ring Z4[u]/uk where uk=0 with =(1+2uk-1) and (3+2uk-1). It is shown that the Gray images of -constacyclic and skew -constacyclic codes over the ring Z4[u]/uk are cyclic, quasi-cyclic, permutation equivalent to a QC code over Z4. Further, the generators of these -constacyclic codes over Z4[u]/uk are obtained.

关键词： constacyclic code Quasi-cyclic code Gray map skew constacyclic code

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

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第4期68卷 2493-2514页

作者： Islam, Habibul Patel, Shikha Prakash, Om Sole, Patrick Indian Inst Technol Patna Dept Math Patna 801106 Bihar India Aix Marseille Univ CNRS Cent Marseille I2M Marseilles France

For a prime p, let F-q2 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-q2 [u(1), u(2), ... , u(r)]/< u(i)(2) - gamma(i)u(i), u(i)u(j) = u(j)u(i) = 0 > where 1 <= i not equal j <= r, gamma(i) is an element of F*(q2), and r >= 1 is an integer. For a unit Lambda = Sigma(r)(i=0) kappa(i)Lambda(i) in the ring A(q,r), we show that a (skew) Lambda-constacyclic code of length n is a direct sum of (skew) Lambda(i)-constacyclic codes of length n over F-q2. 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.

关键词： Non-chain ring constacyclic code skew constacyclic code Gray map Hermitian construction Quantum code

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New Z4 codes from constacyclic codes over a non-chain ring

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COMPUTATIONAL & APPLIED MATHEMATICS 2021年第1期40卷 1-17页

作者： Islam, Habibul Prakash, Om Indian Inst Technol Patna Dept Math Patna 801106 Bihar India

Let Z4 be the ring of integers modulo 4. This paper presents (1+2u+2v+2uv)-constacyclic and skew (1+2u+2v+2uv)-constacyclic codes over the ring Z4+uZ4+vZ4+uvZ4 where u2=u,v2=v,uv=vu. We define three Gray maps and show that the Gray images of (1+2u+2v+2uv)-constacyclic and skew (1+2u+2v+2uv)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Also, we show that cyclic and (1+2u+2v+2uv)-constacyclic codes of odd length are principally generated. As an application, several new quaternary linear codes from the Gray images of (1+2u+2v+2uv)-constacyclic codes are obtained.

关键词： constacyclic code Gray map Quaternary code Quasi-cyclic code skew polynomial ring skew constacyclic code 94B05 94B15 94B35 94B60

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On left quaternion codes

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2016年第10期93卷 1629-1649页

作者： Cao, Yuan Cao, Yonglin Fu, Fang-Wei Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Shandong Univ Technol Sch Sci Zibo 255091 Shandong Peoples R China Nankai Univ Chern Inst Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China

Let F-q be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group Q(n) by the presentation: Q(n) = where n is even and satisfies gcd(n, q) = 1. Left id... 详细信息

Let F-q be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group Q(n) by the presentation: Q(n) = < x, y vertical bar x(n) = 1, y(2) = x(n/2), yxy(-1) = x(-1)> where n is even and satisfies gcd(n, q) = 1. Left ideals of the group algebra F(q)Q(n) are called left quaternion codes over F-q of length 2n, and abbreviated as left Q(n)-codes. In this paper, a system theory for left Q(n)-codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left Q(n)-code is a direct sum of concatenated codes with the inner code A(i) and the outer code C-i, where A(i) is a minimal self-reciprocal cyclic code over F-q of length n and C-i is a skew constacyclic code of length 2 over an extension field or an extension principal ideal ring of F-q. Then we give explicit expressions for outer codes in the concatenated codes, and present the dual code for any left Q(n)-code precisely. Moreover, all distinct left Q(12)-codes over F-5 and all distinct left Q(20)-codes over F-3 are presented, respectively.

关键词： left quaternion code concatenated structure skew constacyclic code cyclic code dual code

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skew cyclic and skew constacyclic codes over the ring F_p

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ASIAN-EUROPEAN JOURNAL OF MATHEMATICS 2019年第5期12卷

作者： Bag, Tushar Upadhyay, Ashish K. Indian Inst Technol Patna Dept Math Patna 801103 Bihar India

In this paper, we study skew cyclic and skew constacyclic codes over the ring F-p + u(1)F(p) + . . . + u(2m)F(p), where u(i)(2) = u(i) and u(i)u(j) = 0, for i, j = 1, . . . , 2m and i not equal j. We have introduced s... 详细信息

关键词： Cyclic code skew cyclic code Gray map quasi-cyclic code skew constacyclic code

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