Search

There is a non-unital ring I of order 4 defined by generators and relations as I = 〈 a , b | 2 a = 2 b = 0 , a 2 = b , a b = 0 〉. In this paper, we present special constructions of linear codes over I from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in I are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given. [ABSTRACT FROM AUTHOR]

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H = 〈 a , b ∣ 2 a = 2 b = 0 , a 2 = 0 , b 2 = b , a b = b a = 0 〉. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2 n over F 2 is studied. [ABSTRACT FROM AUTHOR]

Hulls of linear codes are widely studied due to their good properties and wide applications. Let n = q m - 1 r and C be an [n, k] cyclic code over F q , where r | q - 1 . In this paper, we present several necessary and sufficient conditions for BCH codes of length n that have k - 1 or k ⊥ - 1 dimensional hulls, where k ⊥ is the dimension of C ⊥ . Further, we give the parameters of several families of self-orthogonal codes that arise as hulls of BCH codes. We obtain many optimal codes. [ABSTRACT FROM AUTHOR]

A q-ary linear [n,k,n − k − (i − 1)] code ℒ is known as AiMDS code. If d = n − k + 1, the code is called a maximum distance separable code. MDS codes are known to be useful in the secret-sharing schemes and coding theory. In this paper, we classify all binary self-orthogonal A3MDS codes. We have introduced a subclass of A3MDS codes, called N3STMDS codes, to form a special type of secret-sharing scheme where for any participant, there exist (k − 1) other participants who can collectively reveal the secret by combining their shares and there exists another set of (k + 2) participants including the given participant, who cannot reveal the secret. [ABSTRACT FROM AUTHOR]

In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings Ip = (a, b | pa = pb = 0, a² = b, ab = 0), where p is an odd prime. We also give a classification of the three said classes of codes over Ip where p = 3; 5; and 7, with lengths up to 3. [ABSTRACT FROM AUTHOR]

Let E and I denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as E = (a, b | 2a = 2b = 0, a² = a, b² = b, ab = a, ba = b) and I = (a, b | 2a = 2b = 0, a2 = b, ab = 0). Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings E and I for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of I, and 4 in the case of E. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type IV codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over E and I. [ABSTRACT FROM AUTHOR]

In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings $ {{\mathit {I}_p}} = \left < a, b | pa = pb = 0, a^2 = b, ab = 0 \right > $, where $ p $ is an odd prime. We also give a classification of the three said classes of codes over $ {{\mathit {I}_p}} $ where $ p = 3, 5, $ and $ 7 $, with lengths up to $ 3 $.

Let $ E $ and $ I $ denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as $ E = \langle a, b \mid 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle $ and $ I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle $. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings $ E $ and $ I $ for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of $ I $, and 4 in the case of $ E $. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over $ E $ and $ I $.

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

The authors study the binary codes spanned by the adjacency matrices of the strongly regular graphs (SRGs) on at most two hundred vertices whose existence is unknown. The authors show that in length less than one hundred they cannot be cyclic, except for the exceptions of the SRGs of parameters (85, 42, 20, 21) and (96, 60, 38, 36). In particular, the adjacency code of a (85,42, 20, 21) is the zero-sum code. In the range [100, 200] the authors find 29 SRGs that could possibly have a cyclic adjacency code. [ABSTRACT FROM AUTHOR]

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 22 new codes that improve the minimum distances of best known linear codes (BKLCs). [ABSTRACT FROM AUTHOR]

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 33 new record breaking linear codes over GF(2), GF(3) and GF(5) produced from this method. [ABSTRACT FROM AUTHOR]

Kim and Ohk (2022) showed that binary self-orthogonal [n, k] codes for various dimensions k can be useful for the construction of DNA codes based on quasi self-dual codes over a noncommutative nonunital ring E with four elements. However, there is few classification of binary self-orthogonal codes with dimension ≥ 6 . In this paper, we complete the classification of binary self-orthogonal codes of lengths from 16 to 20 with dimension ≥ 6 . Hence, our classification extends a result of Kim and Ohk (2022). [ABSTRACT FROM AUTHOR]

In this paper, we first construct two infinite families of new two-weight codes over Z 2 m with respect to homogeneous metric and Lee metric by their generator matrices, which generalizes the results in Shi et al (Des Codes Cryptogr 88(3):1–13, 2020) from two different directions. We construct some optimal codes over Z 2 m and prove all codes in one of these two families are self-orthogonal. Finally, we determine the linearity of the Gray images of the codes we constructed for Lee metric completely. [ABSTRACT FROM AUTHOR]

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described in Tonchev (J Combinat Theory Ser A 52:197-205, 1989) in order to obtain self-orthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size is odd and block intersection numbers are even described in Crnković (Adv Math Commun 12:607–628, 2018). Also, we generalize mentioned construction in order to obtain self-orthogonal codes over an arbitrary field. We construct weakly self-orthogonal designs invariant under an action of Mathieu group M 11 and, from them, binary self-orthogonal codes. [ABSTRACT FROM AUTHOR]

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as I = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0 ⟩. We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to n = 6. [ABSTRACT FROM AUTHOR]

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]

A class of self-dual quasi-abelian codes of index 2 over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in $F$. Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good. [ABSTRACT FROM AUTHOR]

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.

There is a local ring E of order 4 , without identity for the multiplication, defined by generators and relations as E = 〈 a , b | 2 a = 2 b = 0 , a 2 = a , b 2 = b , a b = a , b a = b 〉. We study a recursive construction of self-orthogonal codes over E. We classify, up to permutation equivalence, self-orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 1 2. In particular, we classify Type IV codes (QSD codes with even weights) up to n = 1 2. [ABSTRACT FROM AUTHOR]

In this paper, let α + u β be a unit in F p m + u F p m (u 2 = 0) , where p is an odd prime, m is a positive integer and β ≠ 0 . With the help of decomposition of the binomial x 8 - α 0 into a product of irreducible coprime polynomials and the ring (F p m + u F p m) [ x ] x 8 p s - (α + u β) is a principal ideal ring, we give the complete description of all (α + u β) -constacyclic codes of length 8 p s over the finite commutative chain ring F p m + u F p m (u 2 = 0) in terms of their generator polynomials, where α 0 p s = α . We also find out the number of codewords in each of these constacyclic codes. Besides illustrating our results with examples, we determine duals of constacyclic codes, and as an application, we determine the self-dual, self-orthogonal, dual-containing, and linear complimentary-dual (α + u β) -constacyclic codes of length 8 p s over F p m + u F p m (u 2 = 0) . Also, we determine the RT (Rosenbloom–Tsfasman) distances, RT weight distributions, and Hamming distances of such constacyclic 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]

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q 2 -ary linear codes. This result can be generalized to q 2 m -ary linear codes, m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q ≠ 2 , improving others in the literature. [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]

Copyright of Revista Colombiana de Matemáticas is the property of Universidad Nacional de Colombia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

Let F q be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual (LCD) codes and self-orthogonal codes in the finite dihedral group algebras F q [ D 2 n ] . Some numerical examples are also presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

Self-orthogonal codes and self-dual codes, on the one hand, and matrix-product codes, on the other, form important and sought-after classes of linear codes. Combining the two constructions would be advantageous. Adding to this combination the relaxation of the underlying algebraic structures to be commutative rings instead of fields would be even more advantageous. The current article paves a path in this direction. The authors study the problem of self-orthogonality and self-duality of matrix-product codes over a commutative ring with identity. Some methods as well as special matrices are introduced for the construction of such codes. A characterization of such codes in some cases is also given. Some concrete examples as well as applications to torsion codes are presented. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, D 2 n , r a generalized dihedral group with gcd (2 n , q) = 1 , and F q [ D 2 n , r ] a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of F q [ D 2 n , r ] is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in F q [ D 2 n , r ] are precisely described and counted. Some numerical examples are also presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a mass formula for Euclidean and Hermitian self-orthogonal codes over the finite ring Fq + uFq, where Fq is the finite field of order q and u² = 0. We also establish a mass formula for Euclidean self-orthogonal codes over the finite ring Fq + uFq + u2Fq, with u³ = 0 and characteristic of Fq is odd. These mass formulas are used to give a classification of Euclidean and Hermitian self-orthogonal codes over F2 + uF2 and F3 + uF3 of small lengths. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈ui2$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from RkntoFpm2kn$\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6n/2 over these rings (called here quasi self-dual codes or QSD) till the length n=8. To any such code is attached canonically a Ze-code, which, when self-dual, produces an unimodular lattice by Construction A. [ABSTRACT FROM AUTHOR]

There is a local ring H of order 4, without identity for the multiplication, defined by generators and relations as H =(a, b I 2a = 2b = 0, a2 = 0, b2 = b, ab = ba = 0). We classify self orthogonal codes of length n and size 2„ (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify quasi Type IV codes (a subclass of Type IV codes, viz QSD codes with even weights) up to n = 6. [ABSTRACT FROM AUTHOR]

In this paper, we study self-dual codes over finite fields using tools from algebraic function fields in one variable. An algebraic geometry code of length $n$ is defined using two divisors $G$ and $D=P_{1}+\cdots +P_{n}$. We characterize self-orthogonality of the genus zero code in terms of the divisors $G$ , $D$ and the value of a well-chosen derivative polynomial at points $(P_{i})_{1\le i\le n}$. We explore the existence problem of MDS self-dual codes in the odd characteristic case, and we explicitly construct families of new MDS self-dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the equivalence of quantum stabilizer codes via symplectic isometries of stabilizer codes. We define monomially and symplectically equivalent stabilizer codes and determine how different the two notions can be. Further, we show that monomial maps correspond to local Clifford operators. We relate the latter with the LU-LC Conjecture. [ABSTRACT FROM AUTHOR]

Recently certain few-Lee weight additive codes over some mixed alphabet rings with optimal or minimal Gray images are constructed using suitable defining sets. Inspired by these works, we construct two special classes of linear codes over Z 2 [ u ] with u 2 = 0 by using the mixed alphabet ring Z 2 Z 2 [ u ] and employing simplicial complexes having one or more maximal elements. We show that these codes have few-Lee weights by determining their Lee weight distributions. As applications, we are getting the 1st order binary Reed-Muller codes of length 2 3 m − 2 (where m > 1) and some infinite families of binary self-orthogonal minimal codes via Gray map. [ABSTRACT FROM AUTHOR]

In this paper we consider stabilizer codes over local Frobenius rings. Firstly, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems. [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]

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring ℤ p m , where p is a prime number. A generating polynomial of cyclic self-orthogonal codes over ℤ p m is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over ℤ p m . Finally, we determine the number of the above codes with length n over ℤ p m for any (p , n) = 1. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq.6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over ℤ p m . [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for q>3 there exists a q-ary [n, k] Euclidean LCD MDS code, where 0≤k≤n≤q+1, or, q=2m, n=q+2 and k=3orq-1. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes. [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]

The explicit expressions of generator polynomials of cyclic codes of length over finite fields are obtained. The coefficients of these generator polynomials and check polynomials are obtained through modular Lucas sequences. Further, using these polynomials, self-dual, reversible and self-orthogonal cyclic codes of length are classified. [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]