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In this paper, we consider S-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new S-chain search method. As computational results, 18 S-chains with large distances are obtained, and many good quantum codes can be derived from those S-chains by Steane construction, some of which improve the previous results.

Let R r = F q + v 1 F F q + ··· + v r F q , where q is the power of prime, v i 2 = v i , v i v j = v j v i = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring R r is studied and a Gray map ϕ from R r n to F q (r +1) n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determined. As an application, we obtain many new better quantum codes from dual-containing λ-constacyclic codes over R r , for r = 1, 2, 3, that improve the known existing quantum codes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, q = p m , R 1 = F q + u F q + v F q + u v F q with u 2 = 1 , v 2 = 1 , u v = v u and R 2 = F q [ u , v , w ] / ⟨ u 2 - 1 , v 2 - 1 , w 2 - 1 , u v - v u , v w - w v , w u - u w ⟩ with u 2 = 1 , v 2 = 1 , w 2 = 1 , u v = v u , v w = w v , w u = u w . In this paper, F q R 1 R 2 -cyclic codes are introduced. We construct quantum error-correcting codes from F q R 1 R 2 -cyclic codes and introduced a Gray map to find new and better quantum error-correcting codes than previously known quantum error-correcting codes over F q . [ABSTRACT FROM AUTHOR]

A new class of binary quantum codes from cyclic codes over Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ , u 4 = 0 , is introduced. The generator polynomials of Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes of length (r, s) are obtained through the factorization of x r - 1 and x s - 1 into pairwise coprime monic polynomials over Z 2 , where r and s are odd positive integers. A minimal spanning set for these codes is obtained. Under some restricted conditions, the structure of the duals of Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes is also determined. Necessary and sufficient conditions for a Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic code of this restricted class to contain its dual or to be self-orthogonal are obtained. A new Gray map is defined, and the binary quantum codes are obtained by using the Calderbank-Shor-Steane construction on self-orthogonal or dual containing Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes. Some examples of binary quantum codes with good parameters constructed from Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes are given. [ABSTRACT FROM AUTHOR]

Let Rm = Fq[y]/[ym − 1], where m | q − 1. In this paper, we obtain the structure of linear and cyclic codes over Rm. Also, we introduce a preserving-orthogonality Gray map from Rm to Fmq. Among the main results, we obtain the exact structure of self-orthogonal cyclic codes over Rm to introduce parameters of quantum codes from cyclic codes over Rm. [ABSTRACT FROM AUTHOR]

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In this paper, we propose a construction of q-ary quantum codes from Hermitian self-orthogonal cyclic codes over the finite chain ring R = F q 2 [ u ] / ⟨ u k ⟩ , with u k = 0 , where F q 2 is a finite field with q 2 elements, q = p m , p a prime. A Gray map from R to F q 2 k is defined. Some characterizations of cyclic codes over R have been given in terms of their different types of generators. The structure of their dual codes has also been determined, and a necessary and sufficient condition for these codes to be self-orthogonal is presented. The construction of quantum codes is derived by applying Hermitian construction to the Gray images of self-orthogonal cyclic codes over R. From this construction, we have been able to obtain some new quantum codes with better parameters than some presently known comparable best codes available in the literature. Some of these codes have been given as examples, and the others have been given in the form of two tables. [ABSTRACT FROM AUTHOR]

In this paper, quantum codes over Fp from cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉, where u2 = 1, v3 = v, uv = vu and p is an odd prime have been studied. We give the structure of cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 and obtain quantum codes over Fp using self-orthogonal property of these classes of codes. Moreover, by using decomposing method, the parameters of the associated quantum code have been determined. [ABSTRACT FROM AUTHOR]

In the paper, we consider a finite non-chain commutative ring , where . We mainly study the construction of quantum codes from cyclic codes over . For this, we obtained self-orthogonal codes over as gray images of linear and cyclic codes over , then these codes over correspond to a cyclic code over of odd length used to determine the parameters of the quantum codes. [ABSTRACT FROM AUTHOR]

We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them. [ABSTRACT FROM AUTHOR]

In this paper, we study quantum codes over $$F_q$$ from cyclic codes over $$F_q+uF_q+vF_q+uvF_q,$$ where $$u^2=u,~v^2=v,~uv=vu,~q=p^m$$ , and p is an odd prime. We give the structure of cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ and obtain self-orthogonal codes over $$F_q$$ as Gray images of linear and cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ . In particular, we decompose a cyclic code over $$F_q+uF_q+vF_q+uvF_q$$ into four cyclic codes over $$F_q$$ to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over \BBF 4 are used in the construction of many asymmetric quantum codes over \BBF 4. [ABSTRACT FROM PUBLISHER]

We give a construction for quantum codes from linear and cyclic codes over $\mathbb{F}_4 + u\mathbb{F}_4$. We derive Hermitian self-orthogonal codes over $\mathbb{F}_4$ as Gray images of linear and cyclic codes over $\mathbb{F}_4 + u\mathbb{F}_4$. In particular, we use two binary codes associated with a cyclic code over $\mathbb{F}_4 + u\mathbb{F}_4$ of odd length to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper. [ABSTRACT FROM AUTHOR]

Several methods for classifying self-orthogonal codes up to equivalence are presented. These methods are used to classify self-orthogonal codes with largest possible minimum distance over the fields $\mathbb{F}-3$ and $\mathbb{F}-4$ for lengths $n \leq 29$ and small dimensions (up to 6). Some properties of the classified codes are also presented. In particular, an extensive collection of quantum error-correcting codes is obtained. [ABSTRACT FROM AUTHOR]