Self-orthogonal Codes Research Articles

The images of constacyclic codes and new quantum codes

Let q be a prime power and $$m\ge 2$$ be a positive integer. A sufficient condition for the $$q^2$$-ary images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$ to be Hermitian self-orthogonal is presented. Hermitian self-orthogonal codes over $${\mathbb {F}}_{q^{2}}$$ are obtained as the images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$. Two classes of quantum codes are derived by employing the Hermitian construction. The construction produces quantum codes with better parameters than the previously known ones.

Binary LCD Codes and Self-Orthogonal Codes via Simplicial Complexes

Due to some practical applications, linear complementary dual (LCD) codes and self-orthogonal codes have attracted wide attention in recent years. In this paper, we use simplicial complexes for construction of an infinite family of binary LCD codes and two infinite families of binary self-orthogonal codes. Moreover, we explicitly determine the weight distributions of these codes. We obtain binary LCD codes which have minimum weights two or three, and we also find some self-orthogonal codes meeting the Griesmer bound. As examples, we also present some (almost) {\it optimal} binary self-orthogonal codes and LCD {\it distance optimal} codes.

On Euclidean self-dual codes and isometry codes

In this paper, we provide new methods and algorithms to construct Euclidean self-dual codes over large finite fields. With the existence of a dual basis, we study dual preserving linear maps, and as an application, we use them to construct self-orthogonal codes over small finite prime fields using the method of concatenation. Many new optimal self-orthogonal and self-dual codes are obtained.

Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

On Reversibility and Self-Duality for Some Classes of Quasi-Cyclic Codes

In this work, we study two classes of quasi-cyclic (QC) codes and examine how several properties can be combined into the codes of these classes. We start with the class of QC codes generated by diagonal generator polynomial matrices; a QC code in this class is a direct sum of cyclic codes. Then we move on to the class of QC codes of index 2; various binary codes with good parameters are found in this class. In each class, we describe the generator polynomial matrices of reversible codes, self-orthogonal codes, and self-dual codes. Hence, we demonstrate how such properties can be merged in codes of these classes. Particularly for QC codes of index 2, we prove a necessary and sufficient condition for the self-orthogonality of reversible codes. Then we show that reversible QC codes of index 2 are self-dual under the same conditions in which self-dual codes are reversible. We clarify that self-orthogonal reversible QC codes of index 2 over $\mathbb {F}_{q}$ exist for even and odd $q$ , however self-dual reversible codes exist only for even $q$ . Theoretical results are reinforced by several numerical examples. Computer search is used to present some self-dual reversible QC codes of index 2 that have the best known parameters as linear codes. Finally, we highlight the class of 1-generator binary QC codes of index 2 by exploring many self-dual reversible codes that achieve the upper bound on the minimum distance for their parameters.

Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs

<p style='text-indent:20px;'>In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain conditions these orbit matrices yield self-orthogonal codes over finite fields <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a prime power and over finite rings <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}_m $\end{document}</tex-math></inline-formula>. As a case study, we construct codes from orbit matrices of Seidel, Laplacian and signless Laplacian matrices of strongly regular graphs. In particular, we construct self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of the Higman-Sims and McLaughlin graphs.

Quantum Codes from Constacyclic Codes over Polynomial Residue Rings

Sufficient and necessary conditions for Hermitian (1 + λu)-constacyclic self-orthogonal codes over Fpm +uFpm are obtained, where λ is a unit of Fpm + uFpm. Based on this, a new method for the construction of pm-ary quantum codes from the obtained Hermitian (1+ λu) constacyclic self-orthogonal codes over Fpm+uFpm is given. As an example, [[pm - 1; pm - 2d+ 1, d]]pm quantum MDS codes are constructed for 1 ≤ d ≤ (pm +1)/2 and p ≠ 2, as well as some other quantum codes.

New extremal Type II ℤ4-codes of length 32 obtained from Hadamard matrices

For every Hadamard design with parameters [Formula: see text]-[Formula: see text] having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters [Formula: see text]-[Formula: see text]. Moreover, for the case [Formula: see text] we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II [Formula: see text]-codes of length 32. The constructed codes are the first examples of extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text], [Formula: see text], whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text].

Extended maximal self-orthogonal codes

Self-dual and maximal self-orthogonal codes over [Formula: see text], where [Formula: see text] is even or [Formula: see text](mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator [Formula: see text] of a self-dual code [Formula: see text] is given in terms of a maximal self-orthogonal code [Formula: see text], where [Formula: see text] is obtained by the extension of [Formula: see text].

Extending self-orthogonal codes

In this short note we give an exact count for the number of self-dual codes over a finite field $\mathbb F_q$ of odd characteristic containing a given self-orthogonal code. This generalizes an analogous result of MacWilliams, Sloane, and Thompson over the field $\mathbb F_2$ to arbitrary odd finite fields $\mathbb F_q$.

On Constacyclic Codes over $$\boldsymbol{Z}_{\boldsymbol{p}_{1}\boldsymbol{p}_{2}\cdots\boldsymbol{p}_{\boldsymbol{t}}}$$

Let t ≥ 2 be an integer, and let p1, ⋯, pt be distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring $${Z_{{p_1}{p_2} \cdots {p_t}}}$$ and the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over $${Z_{{p_1}{p_2} \cdots {p_t}}}$$ .

Binary LCD Codes and Self-Orthogonal Codes From a Generic Construction

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.

Some classes of LCD codes and self-orthogonal codes over finite fields

Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general $ p $-ary case, where $ p $ is an odd prime. Based on the extended construction, several classes of $ p $-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

New Non-binary Quantum Codes Over &amp;lt;i&amp;gt;F&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;+uF&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;+vF&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;+uvF&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;

Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.

The classification of self-orthogonal codes of length 42

Self-orthogonal codes play an important role in constructing quantum-error-correcting codes. In this paper, we prove that if quasi-symmetric 2-(41, 9, 9) design exists, then it arises from self-orthogonal and self-complementary [Formula: see text] codes with dual distance of at least 5. Moreover, we emphasize the enumeration of inequivalent doubly-even codes with the needed dual distance and an automorphism of order 7. This is found to be precisely 8.

Cyclic self-orthogonal codes over finite chain ring

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. 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 [Formula: see text].

On cyclic and negacyclic codes of length 8ℓmpn over finite field

Let [Formula: see text] be finite field with [Formula: see text] elements, where [Formula: see text] be power of an odd prime [Formula: see text] In this paper, we determine generator polynomials of all cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] are distinct odd primes and [Formula: see text] are positive integers. We also determine all self-dual, self-orthogonal, complementary-dual cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text]

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs.

On the list decodability of self-orthogonal rank-metric codes

Guruswami and Resch proved that a random Fq-linear rank-metric code is list decodable with list decoding radius attaining the Gilbert–Varshamov bound [8]. Furthermore, in Hamming metric, random linear self-orthogonal codes can be list decoded up to the Gilbert–Varshamov bound with polynomial list size [11]. Motivated by these two results and the potential applications of self-orthogonal rank-metric codes in network coding and cryptography [20], [18] and [5], we focus on investigating their list decodability. In this paper, we prove that with high probability, a random Fq-linear self-orthogonal rank-metric code over Fqn×m can be list decoded up to the Gilbert–Varshamov bound with polynomial list size. In addition, we show that an Fqm-linear self-orthogonal rank-metric code of rate up to the Gilbert–Varshamov bound with exponential list size.

Quantum stabilizer codes construction from Hermitian self-orthogonal codes over GF(4)

In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.