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We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum. [ABSTRACT FROM AUTHOR]

We give a new technique for constructing self-dual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct self-dual codes over finite commutative Frobenius rings of characteristic 2. We give and prove the necessary conditions needed for the technique to produce self-dual codes. We also establish the connection between self-dual codes generated by the new technique and units in group rings. Using the construction together with the building-up construction, we obtain new extremal binary self-dual codes of lengths 64, 66 and 68 and new best known binary self-dual codes of length 80. [ABSTRACT FROM AUTHOR]

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when p = 2 , which gives an affirmative answer to Heng's conjecture. [ABSTRACT FROM AUTHOR]

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach. [ABSTRACT FROM AUTHOR]

Let S = ℤp[u, v]/〈u2, v2, uv − uv〉 be a semi-local ring, where p is a prime number. In the present article, we determine the generating sets of S and use them to construct the structures of ℤpS-additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤpS-additive cyclic and constacyclic codes are also determined. These codes are identified as S[y]-submodules of the ring Sβ1, β2 = ℤp[y]/〈yβ1 − 1〉 × S[y]/〈yβ2 − 1〉. Some results that represent the relationship between the minimal polynomials of ℤpS-additive cyclic codes and their duals have been obtained. Furthermore, optimal ℤpS-additive codes and maximum distance separable codes have been evaluated (see Table 1). Finally, we use MAGMA software to find the parameters of Optimal and MDSS codes.

In this note, we complete a classification of ternary extremal four-negacirculant self-dual codes of lengths 40, 44, 48, 52 and 60. [ABSTRACT FROM AUTHOR]

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [ABSTRACT FROM AUTHOR]

Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained by the defining sets over finite fields of odd characteristic. The parameters and weight distributions of linear codes are determined by the additive characters, multiplicative characters and Gauss sums. Further, most of linear codes obtained are minimal, which can be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper, δ -dual codes over finite commutative semi-simple rings are defined as a generalization of dual codes over finite commutative semi-simple rings. Some properties of δ -dual codes are given. We present necessary and sufficient conditions for a λ -constacyclic code of length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. We also study the δ -dual of skew Θ - λ -constacyclic codes over finite commutative semi-simple rings. Among others, we also give necessary and sufficient conditions for a skew Θ - λ -constacyclic code of any length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. [ABSTRACT FROM AUTHOR]

Let p be an odd prime with p ≠ 5 . In this paper, we first provide the structures of repeated-root cyclic codes of length 10 p s over finite fields F p m . We then give two methods of constructing good quantum error-correcting (QEC) codes from repeated-root cyclic codes of length 10 p s over finite fields F p m . By means of the dimensions of repeated-root cyclic codes of length 10 p s over finite fields F p m , we exhibit an effective manner for constructing new EAQEC codes. We show that the usage of these methods brings us many good QEC and EQAEC codes having these advantages: (1) the parameters of our QEC and EQAEC codes are different from all the previous constructions; (2) for repeated-root cyclic codes, our methods allows for easily calculating the dimensions of QEC and EQAEC codes, and the numbers c of pre-shared maximally entangled states of EAQEC codes. [ABSTRACT FROM AUTHOR]

A locally repairable code (LRC) is a code that can recover any symbol of a codeword by reading at most r other symbols, denoted by r -LRC. In this paper, we study binary and ternary linear LRCs with disjoint repair groups and minimum distance d = 6. Using the intersection subspaces technique, we explicitly construct dimensional optimal LRCs. First, based on the intersection subspaces constructed by t -spread, a construction of binary LRCs is designed. Particularly, a class of binary linear LRCs with r = 11 is optimal in terms of achieving a sphere-packing type upper bound. Next, by using the Kronecker product of two matrices, two classes of dimensional optimal ternary LRCs with small locality (r = 3, 5) are presented. Compared to previous results, our construction is more flexible regarding code parameters. Finally, we also discuss the parameters of a code obtained by applying a shortening operation to our LRCs. We show that these shortened LRCs are also k -optimal. [ABSTRACT FROM AUTHOR]

In this paper we give three constructions of cyclic self-orthogonal codes over Z 2 k , for k ≥ 3 , from Boolean functions on n variables. The first construction for each k, 3 ≤ k ≤ n , yields a self-orthogonal Z 2 k -code of length 2 n + 2 with all Euclidean weights divisible by 2 k + 1. In the remaining two constructions, for each even n and k ≥ 3 , we generate a self-orthogonal Z 2 k -code of length 2 n + 1. All Euclidean weights in the constructed code are divisible by 2 2 k - 1 or 2 k + 1 , depending on which of the two constructions is used. [ABSTRACT FROM AUTHOR]

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve. [ABSTRACT FROM AUTHOR]

The study of aperiodic Hamming correlation (APC) of frequency hopping sequences (FHSs) is a hard problem that has not been paid enough attention in the literature. For low-hit-zone (LHZ) FHSs, the study of APC becomes more difficult. We call them LHZ FHSs under APC (LHZ-APC FHSs). LHZ-APC FHSs are studied for the first time in this paper. First, we establish a bound on the family sizes of LHZ-APC FHS sets. Then we present a method for constructions of LHZ-APC FHS sets based on conventional FHS sets under periodic Hamming correlation (conventional PC FHS sets). By choosing different conventional PC FHS sets, we obtain three classes of LHZ-APC FHS sets whose family sizes are optimal or near optimal according to this new bound. Further, we modify the construction method and get more new LHZ-APC FHS sets with optimal family sizes. [ABSTRACT FROM AUTHOR]

Partial spreads are important in finite geometry and can be used to construct linear codes. From the results in (Des. Codes Cryptogr. 90, 1–15, 2022) by Xia Li, Qin Yue and Deng Tang, we know that if the number of the elements in a partial spread is "big enough", then the corresponding linear code is minimal. This paper used the sufficient condition in (IEEE Trans. Inf. Theory 44(5), 2010–2017, 1998) to prove the minimality of such linear codes. In the present paper, we use the geometric approach to study the minimality of linear codes constructed from partial spreads in all cases. [ABSTRACT FROM AUTHOR]

In this paper, we find a necessary and sufficient condition for multi-twisted Reed-Solomon codes to be MDS. In particular, we introduce a new class of MDS double-twisted Reed-Solomon codes C α , t , h , η with twists t = (1 , 2) and hooks h = (0 , 1) over the finite field F q , providing a non-trivial example over F 16 and enumeration over the finite fields of size up to 17. Moreover, we obtain necessary conditions for the existence of multi-twisted Reed-Solomon codes with small dimensional hull. Consequently, we derive conditions for the existence of MDS multi-twisted Reed-Solomon codes with small dimensional hull. [ABSTRACT FROM AUTHOR]

Let R be the Galois ring of characteristic 4 and cardinality 4 m , where m is a natural number. Let C be a linear code of length n over R and Φ be the Homogeneous Gray map on R n . In this paper, we show that Φ (C) is linear if and only if for every X , Y ∈ C , 2 (X ⊙ Y) ∈ C . Using the generator polynomial of a cyclic code of odd length over R, a necessary and sufficient condition is given which its Gray image is linear. [ABSTRACT FROM AUTHOR]

Let SLAut (F q n) denote the group of all semilinear isometries on F q n , where q = p e is a prime power. In this paper, we investigate some general properties of linear codes associated with the σ duals for σ ∈ SLAut (F q n) . We show that the dimension of the intersection of two linear codes can be determined by generator matrices of such codes and their σ duals. We also show that the dimension of the σ hull of a linear code can be determined by a generator matrix of it or its σ dual. We give a characterization on the σ dual and σ hull of a matrix-product code. We give an explanation for why it is meaningful to extend the ℓ -Galois hulls of matrix-product codes to the σ hulls of matrix-product codes. We also investigate the intersection of a pair of matrix-product codes. Finally, we construct several families of σ dual-containing matrix-product codes, some of which are optimal or almost optimal according to the Database (Grassl 2023). [ABSTRACT FROM AUTHOR]

In this paper, we give constructions of self-orthogonal codes from orbit matrices of Deza graphs, normally regular digraphs and Deza digraphs in terms of a definition given by Wang and Feng. These constructions can also be applied to adjacency matrices of the mentioned graphs. Since a lot of constructions of Deza graphs, normally regular digraphs and Deza digraphs in the sense of Wang and Feng have been known, the methods presented in this paper give us a rich source of matrices that span self-orthogonal codes. [ABSTRACT FROM AUTHOR]

For an odd prime p and a positive integer r, let q = p r . The objective of this article is to study cyclic codes over the ring S = F q [ u , v , w ] / ⟨ u 3 - u , v 2 - v , w 2 - w , u v , v u , u w , w u , v w - w v ⟩ and to construct new and better quantum and LCD codes from them. We give the structure of cyclic codes over the ring S and obtain dual-containing codes over F q as the Gray images of dual-containing cyclic codes over S . Using these dual-containing codes, we obtain quantum codes and determine their parameters using the decomposition of cyclic codes over the ring S . We provide many new and better-than-existing quantum codes. We also give a method to obtain linear complementary dual (LCD) codes over S using the decomposition of cyclic codes over the ring S . We obtain some optimal and best-known linear codes as Gray images of LCD codes over S . [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

Let a and m be positive integers and λ be any unit in GR (p a , m) of the form λ = (σ 0 + p σ 1 + p 2 z) , where σ 0 , σ 1 ∈ T (p , m) \ { 0 } and z ∈ GR (p a , m) . The symbol-pair distance of all such λ -constacyclic codes over GR (p a , m) of length p s are determined. As an application, we identify all maximum distance separable (MDS) λ -constacyclic codes of length p s over GR (p a , m) with respect to the symbol-pair distance. We give numerous examples of newly constructed MDS symbol-pair codes, i.e., new optimal symbol-pair codes with respect to the Singleton bound. [ABSTRACT FROM AUTHOR]

In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups M 11 , M 12 , M 22 , M 23 and M 24 . From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to M 22 , whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce F q 2 -double cyclic codes of length n = r + s , where F q 2 is the Galois field of q 2 elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any F q 2 -double cyclic code. For any F q 2 -double cyclic code C , we will define the Euclidean dual code C ⊥ based on the Euclidean inner product and the Hermitian dual code C ⊥ H based on the Hermitian inner product. We will construct a relationship between C ⊥ and C ⊥ H and then find the generator polynomials for the Hermitian dual code C ⊥ H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F 4 and also examples of optimal quantum codes that were derived from F 4 -double cyclic codes using the Hermitian inner product. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) quantum error-correcting codes (QECCs) and MDS entanglement-assisted QECCs (EAQECCs) play significant roles in quantum information theory. In this paper, we construct several new families of MDS QECCs and MDS EAQECCs by utilizing Hermitian self-orthogonal generalized Reed–Solomon codes. These newly obtained MDS QECCs contain some known classes of MDS QECCs as subclasses and some of them have larger minimum distance. In addition, many q-ary MDS QECCs and MDS EAQECCs in our constructions have length exceeding q + 1 and minimum distance surpassing q 2 + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we propose a construction of quantum codes from Hermitian self-orthogonal matrix product codes over the finite fields. This construction is applied to obtain numerous new quantum codes, and all of them have higher rate than current quantum codes available. [ABSTRACT FROM AUTHOR]

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight. [ABSTRACT FROM AUTHOR]

We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of t-spreads. It is shown that the dimension of such codes is two and the minimum rank distance is at least half of the length. We construct antipodal two-weight rank metric codes from certain MRD codes. A complete classification of such codes is obtained, when the minimum rank distance is equal to half of the length. As a consequence of our construction of two-weight rank metric codes, we get some explicit two-weight Hamming metric codes. [ABSTRACT FROM AUTHOR]

Galindo et al. introduced a class of codes which are obtained by evaluation of polynomials at the roots of a trace map (Galindo et al. in IEEE Trans Inform Theory 65: 2593–2602, 2019). Via subfield subcodes, this construction yields new linear codes with good parameters as well as good resulting quantum codes. Here, we extend this construction to allow evaluation at the roots of any polynomial which splits in the field of evaluation. Our proof relies on Galois-closedness of codes in consideration. Moreover, we introduce a lengthening process that preserves Galois-closed property of restricted evaluation codes. Subfield subcodes of such lengthened codes yield further good linear codes. In total, we obtain 17 linear codes over F 4 and F 5 which improve the best known linear code parameters in Grassl (Bounds on the minimum distance of linear codes and quantum codes, 2022, http://www.codetables.de). Moreover, we give a construction for two linear codes which have the best known parameters according to Grassl (Bounds on the minimum distance of linear codes and quantum codes, 2022, http://www.codetables.de), but for which no construction was known before. [ABSTRACT FROM AUTHOR]

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm. [ABSTRACT FROM AUTHOR]

Simplex and MacDonald codes have received significant attention from researchers since the inception of coding theory. In this work, we present the construction of linear torsion codes for simplex and MacDonald codes over the ring R = R 1 R 2 R 3 . We have introduced a novel family of linear codes over F p . These codes have been extensively examined with respect to their properties, such as code minimality, weight distribution, and their applications in secret sharing schemes. In addition to this investigation, we have discovered that these codes are also applicable to the association schemes of linear torsion codes for simplex and MacDonald codes over. R = R 1 R 2 R 3 . [ABSTRACT FROM AUTHOR]

In this paper, we first give two new methods for constructing self-orthogonal codes from known self-orthogonal codes. On the basis of this, we construct four infinite classes of binary self-orthogonal codes. Moreover, we also determine their weight distributions and the minimum distances of their dual codes. Furthermore, we present a class of optimal linear codes and a class of almost optimal linear codes with respect to the Sphere Packing Bound. [ABSTRACT FROM AUTHOR]