Generalized Reed-Solomon codes form the most prominent class of maximum distance separable (MDS) codes, codes that are optimal in the sense that their minimum distance cannot be improved for a given length and code size. The study of codes that are MDS yet not generalized Reed-Solomon codes, called non-generalized Reed-Solomon MDS codes, started with the work by Roth and Lemple (1989), where the first examples were exhibited. It then gained traction thanks to the work by Beelen et al. (2017), who introduced twisted Reed-Solomon codes, and showed that families of such codes are non-generalized Reed-Solomon MDS codes. Finding non-generalized Reed-Solomon MDS codes is naturally motivated by the classification of MDS codes. In this paper, we provide a generic construction of MDS codes, yielding infinitely many examples. We then explicit families of non-generalized Reed-Solomon MDS codes. Finally we position some of the proposed codes with respect to generalized twisted Reed-Solomon codes, and provide new view points on this family of codes.

The Assmus-Mattson Theorem is a famous and effective way to constructt-designs from supports of codewords of a linear code. By extending this idea, we study thet-designs constructed from supports of subcodes of linear codes. In this paper, we introduce the notion ofr-generalized maximum distance separable codes (r-GMDS codes) and determine thet-designs constructed from supports of subcodes of ar-GMDS code. Also, we show thatt-designs can be constructed from supports of subcodes of a linear code if the automorphism group of this linear code ist-transitive. In particular, we prove thatt-designs can be constructed from supports of subcodes of Reed-Muller codes. Then we show that the number of blocks in thet-design constructed from supports of subcodes of a linear code is related to the subcode support weight distributions of this linear code. Next, we provide some new formulas for the subcode support weight distributions ofr-GMDS codes. After we present a connection betweenl-MDS codes andr-GMDS codes, some new formulas are obtained for the subcode support weight distributions ofl-MDS codes. In particular, we completely determine the subcode support weight distributions of maximum distance separable (MDS) codes and near maximum distance separable (NMDS) codes. As an example, the subcode support weight distributions of the extended ternary Golay code are completely determined.

In this paper, we propose a new method for constructing a class of non-GRS MDS codes. The lengths of these codes can reach up toq+3/2 (for finite fields of odd characteristic) andq+4/2 (for even characteristic), respectively. Owing to their special structure, we can use the Cauchy matrix method to obtain the necessary and sufficient conditions for these codes to be MDS codes and non-GRS MDS codes. Additionally, the inequivalence between these codes and twisted GRS codes is analyzed. Furthermore, we analyze the relationships among several existing classes of codes used for constructing non-GRS MDS codes, propose explicit constructions, and discuss the lengths of non-GRS MDS codes based on these constructions. Finally, we design two efficient algorithms to address two main problems in GRS code research, i.e., determining whether an unknown codeCis a GRS code from its generator matrixG, and recovering the key vectors α andvsuch thatC=GRSn,k(α,v) ifCis indeed a GRS code. A computational complexity comparison of the proposed algorithms (O(nk+n)) with that of the Sidelnikov-Shestakov attack (exceedingO(qk2n+qk3))shows that our methods offer superior computational efficiency.

Abstract The Rankin constant $$\gamma _{n,l}$$ γ n , l measures the largest volume of the densest sublattice of rank l of a lattice $$\Lambda \in {\mathbb {R}}^n$$ Λ ∈ R n over all such lattices of rank n. The Bergé-Martinet constant $$\gamma '_{n,l}$$ γ n , l ′ is a variation that takes into account the dual lattice. Exact values and bounds for both constants are mostly open in general. We consider the case of lattices built from linear codes, and look at bounds on $$\gamma _{n,l}$$ γ n , l and $$\gamma '_{n,l}$$ γ n , l ′ . In particular, we revisit known results for $$n=3,4,5,8$$ n = 3 , 4 , 5 , 8 and give lower and upper bounds for the open cases $$\gamma _{5,2},\gamma _{7,2}$$ γ 5 , 2 , γ 7 , 2 and $$\gamma '_{5,2},\gamma '_{7,2}$$ γ 5 , 2 ′ , γ 7 , 2 ′ .

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes over$\mathbb {F}_{q^{m}}$are generalized Reed-Solomon codes (GRS codes) when$m=1$. In this paper, we give a necessary and sufficient condition for the Galois dual codes of Goppa codes when$m=1$to be Goppa codes with the same locator sets. Furthermore, we show that the Galois hulls of the above codes are still the Goppa codes and determine their Goppa polynomials and dimensions. In particular, we characterize Galois linear complementary dual (LCD), Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such codes. Moreover, we apply these results to EAQECCs.

Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$-linear codes and they meet all Griesmer bounds on the generalized Hamming weights of linear codes. All the subcodes with dimension r of a$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$-simplex code have the same subcode support weight$\frac {q^{k-r}(q^{r}-1)}{q-1}$for$1\leq r\leq k$. In this paper, we construct linear codes meeting the Griesmer bound of the r-generalized Hamming weight, such codes do not meet the Griesmer bound of the j-generalized Hamming weight for$1\leq j\lt r$. Moreover these codes have only few subcode support weights (few-SSW). The weight distributions and the subcode support weight distributions of these distance-optimal codes are determined. Linear codes constructed in this paper are natural generalizations of distance-optimal few-weight codes.

In this paper, we propose three Griesmer type bounds for the minimum Hamming weight of complementary codes of linear codes. Infinite families of complementary codes meeting the three Griesmer type bounds are given to show these bounds are tight. The Griesmer type bounds proposed in this paper are significantly stronger than the classical Griesmer bound for linear codes. As a by-product, we construct some optimal few-weight codes and determine their weight distributions. As an application, Griesmer type bounds for the column distance of convolutional codes are presented. These Griesmer type bounds are stronger than the Singleton bound for convolutional codes.

Matrix product codes are generalizations of some well-known classes of codes, including generalized Reed-Muller codes and repeated-root constacyclic codes. Recently, a bound for the minimum symbol-pair distance of a matrix product code was given by (Luo et al., 2023), leading to the creation of new families of MDS symbol-pair codes. In this paper, we provide lower and upper bounds for the minimum b-symbol distance of matrix product codes, which naturally extends some of the results by (Luo et al., 2023). Examples meeting the bounds are included to illustrate our results. As an initial application of these new bounds, we establish that constacyclic codes (repeated-root or otherwise) meeting specific criteria can indeed be classified as matrix product codes, and present bounds on the minimum b-symbol distance of such constacyclic codes. Additionally, we determine all the minimum b-symbol distances of Reed-Muller codes as a secondary application of the new bounds.

In this paper, firstly, we study decoding of a general class of twisted generalized Reed-Solomon (TGRS) codes and provide a precise characterization of the key equation for TGRS codes and propose a decoding algorithm. Secondly, we further study decoding of almost-MDS TGRS codes and provide a decoding algorithm. These two decoding algorithms are more efficient in terms of performance compared with the decoding algorithms presented in [Sun et al., IEEE-TIT, 2024] and [Sui et al., IEEE-TIT, 2023] respectively. Moreover, these two optimized decoding algorithms can be applied to the decoding of a more general class of twisted Goppa codes.

LetCbe an arbitrary simple-root cyclic code and letGbe the subgroup of Aut(C) (the automorphism group ofC) generated by the multiplier, the cyclic shift and the scalar multiplications. To the best of our knowledge, the subgroupGis the largest subgroup of Aut(C). In this paper, an explicit formula, in some cases an upper bound, for the number of orbits ofGonC\{0} is established. An explicit upper bound on the number of non-zero weights ofCis consequently derived and a necessary and sufficient condition for the codeCmeeting the bound is exhibited. Many examples are presented to show that our new upper bounds are tight and are strictly less than the upper bounds in [Chen and Zhang, IEEE-TIT, 2023]. In addition, for two special classes of cyclic codes, smaller upper bounds on the number of non-zero weights of such codes are obtained by replacingGwith larger subgroups of the automorphism groups of these codes. As a byproduct, our main results suggest a new way to find few-weight cyclic codes.

Function-correcting codes (FCCs) are a class of codes designed to protect the function evaluation of a message against errors whose key advantage is the reduced redundancy. In this paper, we develop the theory of FCCs over symbol-pair read channels. We introduce the notion of function-correcting symbol-pair codes (FCSPCs) and aim to find their optimal redundancy. To this end, we introduce the notion of irregular-pair-distance codes and derive upper and lower bounds on the optimal redundancy in terms of the shortest length of the irregular-pair-distance codes. We then simplify these bounds and employ these general results to specific functions including pair-locally binary functions, pair weight functions and pair weight distribution functions. In addition, we provide some general constructions for FCSPCs. Lastly, by comparison with classical symbol-pair codes, we find that the theory of FCSPCs developed in our paper really reduces the redundancy under the condition that the receiver can recover certain attribute of the message.

Classical error-correcting codes under the Hamming metric are used to correct substitution and erasure errors. Motivated by the limitations of the reading process in high density data storage systems, a new class of codes called symbol-pair (metric) codes was designed to protect against pair errors in symbol-pair read channels. For a given alphabet of size$q$and given values of$n$and$d$with$1\leq d\leq n$, let$A_{p}(n,d,q)$denote the largest possible code size for which there exists a$q$-ary code of length$n$with minimum pair-distance at least$d$. In this paper, new upper and lower bounds on$A_{p}(n,d,q)$are presented. Several examples are included to illustrate our main results; some examples are optimal in the sense that they meet the corresponding bounds, and the rest examples are meant to show that our bounds may perform better than some of the previously known ones in certain cases. In addition, we show that any symbol-pair code over$\mathbb {F}_{q}$can be viewed as a Hamming metric code over$\mathbb {F}_{q^{2}}$with the same parameters. Consequently, the theory of classical codes over$\mathbb {F}_{q^{2}}$can be used directly to symbol-pair codes; in particular, by virtue of this result, some known results can be reobtained immediately.

Let${\mathrm{ RS}}_{k}(\mathbf {a})$be a$k$-dimensional Reed-Solomon (RS) code over$\mathbb {F}_{q}$associated with$\mathbf {a}=(\alpha _{1},\cdots,\alpha _{n})$and let$h=\prod _{i=1}^{n}(z-\alpha _{i})$be a polynomial in variable$z$. In this paper, by expressing${\mathrm{ RS}}_{k}(\mathbf {a})$as an$\mathcal {L}$-construction algebraic geometry code, we completely determine the dimension of the hull${\mathrm{ RS}}_{k}(\mathbf {a})\bigcap {\mathrm{ RS}}_{k}(\mathbf {a})^{\perp} $in terms of the degree of the derivative of$h$and some relevant polynomials. As applications, we explicitly determine the parameters of MDS entanglement-assisted quantum error-correcting codes constructed from RS codes, and all linear complementary dual (resp. self-dual) RS codes are also fully described.

Generalized pair weights of linear codes are generalizations of minimum symbol-pair weights, which were introduced by Liu and Pan (2022) recently. Generalized pair weights can be used to characterize the ability of protecting information in the symbol-pair read wire-tap channels of type II. In this paper, we introduce the notion of generalized $b$ -symbol weights of linear codes over finite fields, which is a generalization of generalized Hamming weights and generalized pair weights. We obtain some basic properties and bounds of generalized $b$ -symbol weights which are called Singleton-like bounds for generalized $b$ -symbol weights. As examples, we calculate the generalized weight matrices for simplex codes and Hamming codes. We provide a necessary and sufficient condition for a linear code to be a $b$ -symbol MDS code by using the generator matrix and the parity check matrix of this linear code. Finally, a necessary and sufficient condition of a linear isomorphism preserving $b$ -symbol weights between two linear codes is obtained. As a corollary, we get the classical MacWilliams extension theorem when $b=1$ .

In this paper, we introduce the notion of generalized pair weights of an$[n, k]$-linear code over the finite field$\mathbb {F}_{q}$and the notion of pair$r$-equiweight codes, where$1\le r\le k-1$. Some basic properties of generalized pair weights of linear codes over finite fields are derived. We obtain a necessary and sufficient condition for an$[n,k]$-linear code to be a pair equiweight code, and we characterize pair$r$-equiweight codes for any$1\le r\le k-1$. A necessary and sufficient condition for a linear isomorphism to preserve pair weights between two linear codes is obtained. At the end of this paper, an application of generalized pair weights of linear codes to symbol-pair read wire-tap channels of type II is introduced.

Locally repairable codes (LRCs) and codes with hierarchical locality (H-LRCs) have a great significance due to their applications in distributed storage systems. Constructing LRCs or H-LRCs achieving the Singleton-type bound is a challenging task and has received much attention. In this paper, we make use of optimal LRCs to construct a family of optimal H-LRCs. For this purpose we first construct a new class of optimal LRCs via generalized Reed-Solomon codes (GRS codes). Next, based on these optimal LRCs we present some new optimal H-LRCs, of which the parameters are more flexible than those given in [1] and [18].

Let p be a prime number and q = psfor a positive integer s. For any positive divisor e of q - 1, we construct infinite families of codes C of size q2mwith few Lee-weight. These codes are defined as trace codes over the ring R = Fq+ uFq, u2= 0. Using Gaussian sums, their Lee weight distributions are provided. In particular, when gcd(e, m) = 1, under the Gray map, the images of all codes in C are of two-weight over the finite field Fq, which meet the Griesmer bound. Moreover, when gcd(e, m) = 2, 3, or 4, all codes in C are of most five-weight codes.

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended.

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over GR(2a, m) of length 2shave been characterized: the ring R2(a, m, -1) = GR(2a, m)[x]/(x2st 1) is a chain ring. Furthermore, these results have been generalized to λ-constacyclic codes for any unit λ of the form 4z - 1, z ∈ GR(2a, m). In this paper, we study more general cases and investigate all cases, where Rp(a, m, y) = GR(pa, m)[x]/(xps - y) is a chain ring. In particular, the necessary and sufficient conditions for the ring Rp(a, m, y) to be a chain ring are obtained. In addition, by using this structure we investigate all y-constacyclic codes over GR(pa, m) when Rp(a, m, y) is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual y-constacyclic codes are also provided. Among others, for any prime p, the structure of Rp(a, m, y) = GR(pa, m)[x]/(xps- y) is used to establish the Hamming and homogeneous distances of y-constacyclic codes.

We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes.