In this paper, we show that, algebraically, the intersection of two GRS codes is a direct sum of some like-generalized Reed-Solomon codes, and that the dimension of such code can be given via the dimensions of the GRS codes and the degrees of some relevant polynomials. We also provide a necessary and sufficient condition for this intersection to be a GRS code. Our results naturally extend the main results in [11, 16, 19, 23]. Particularly, we deterministically construct two GRS codes with given code length, dimensions, and intersection dimension. As an application of our main results, we derive the algebraic structure of the hull of a GRS code and exhibit a necessary and sufficient condition for the hull to be a GRS code. In addition, we discuss when a GRS code is self-orthogonal or dual-containing and when the hull of an RS code is again an RS code. Finally, as an application, we resolve the problem of explicit construction of MDS EAQECCs from classical codes forn≤q. Several examples are included to illustrate our results.

Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce bounds on the parameters of quantum LRCs from bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum CSS codes, we offer the first construction of several families of optimal pure quantum LRCs.

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.

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.

Cyclically permutable codes (CPCs) have found important applications in many communication systems, such as the multiple access collision channel without feedback, frequency-hopping spread spectrum communication channels and the digital watermarking systems. In this paper, by introducing a new method we completely settle the problem of constructing a CPC with the largest possible code size derived from a given simple-root cyclic code. The contribution of this paper is twofold. First, we present a new enumerative formula for the code size of such CPC with all the terms being positive integers, contrasting to the previously known ones given in [1], [20], [22], [24] which involve the Möbius function. Second, we provide an algebraic and systematic method to produce such a CPC. Several examples are also included to illustrate our main results.

A cross-bifix-free code of lengthnover Zqis a non-empty subset of Znqsuch that the prefix set of each codeword is disjoint from the suffix set of every codeword. To achieve good performance in communication systems, it is desirable to construct cross-bifix-free codes with large size. Recently, Wang and Wang generalized the classical cross-bifix-free codes presented by Levenshtein, Gilbert and Cheeet al. by constructing a new family of cross-bifix-free codesS(k)I,J(n). The codeS(k)I,J(n) is nearly optimal in terms of its size and non-expandable ifk=n- 1 or 1 ≤kn/2. There are three major ingredients in this paper. The first is to improve the results in [Cheeet al., IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022] in which we prove that the codeS(k)I,J(n) is non-expandable if and only ifk=n- 1 or 1 ≤kn/2. The second ingredient contributes to a new family of cross-bifix-free codesU(t)I,J(n). This new code enables us to construct non-expandable cross-bifix-free codesS(k)I,J(n) ᑌU(t)I,J(n) wheneverS(k)I,J(n) is expandable. The union ofU(t)I,J(n) andS(k)I,J(n) enlarges the size ofS(k)I,J(n). Finally, we give an explicit formula for the size ofS(k)I,J(n) ᑌU(t)I,J(n).

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.

Let$\mathcal {C}$be a simple-root cyclic code and let$\mathcal {G}$be the subgroup of the automorphism group of$\mathcal {C}$generated by the cyclic shift of$\mathcal {C}$and the scalar multiplications of$\mathcal {C}$. In this paper, we find an explicit formula for the number of orbits of$\mathcal {G}$on$\mathcal {C}\setminus \{\mathbf {0}\}$. Consequently, an explicit upper bound on the number of non-zero weights of$\mathcal {C}$is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. Several reducible and irreducible cyclic codes meeting the bound are presented, revealing that our bound is tight. In particular, we find that some infinite families of irreducible cyclic codes constructed in (Ding, 2009) meet our bound; we then conclude that such known codes enjoy an additional property that any two codewords with the same weight belong to the same$\mathcal {G}$-orbit, a fact that may not have been known before. Our main result improves and generalizes some of the results in (Shi et al., 2019).

Goppa, in the 1970s, discovered the relation between algebraic geometry and codes, which led to the family of Goppa codes. As one of the most interesting subclasses of linear codes, the family of Goppa codes is often chosen as a key in the McEliece cryptosystem. Knowledge of the number of inequivalent binary Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. Let$n\geq 5$be an odd prime number, let$q=2^{n}$and let$r\geq 3$be a positive integer satisfying$\gcd (r,n)=1$. The purpose of this paper is to establish an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length$q+1$and degree$r$. A potential mathematical object for this purpose is to count the number of orbits of the projective semi-linear group${\mathrm{ PGL}}_{2}(\mathbb {F}_{q})\rtimes {\mathrm{ Gal}}(\mathbb {F}_{q^{r}}/\mathbb {F}_{2})$on the set$\mathcal {I}_{r}$of all monic irreducible polynomials of degree$r$over the finite field$\mathbb {F}_{q}$. An explicit formula for the number of orbits of${\mathrm{ PGL}}_{2}(\mathbb {F}_{q})\rtimes {\mathrm{ Gal}}(\mathbb {F}_{q^{r}}/\mathbb {F}_{2})$on$\mathcal {I}_{r}$is given, and consequently, an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length$q+1$and degree$r$is derived. Our main result naturally contains the main results of Ryan (IEEE-TIT 2015), Huang and Yue (IEEE-TIT, 2022) and, Chen and Zhang (IEEE-TIT, 2022), which considered the cases$r=4$,$r=6$and$\gcd (r,q^{3}-q)=1$respectively.

Locally repairable codes with hierarchical locality (H-LRCs) are designed to correct different numbers of erasures, which play a crucial role in large-scale distributed storage systems. In this paper, we construct three classes of$q$-ary optimal H-LRCs by employing matrix product codes, concatenated codes and cyclic codes, respectively. The first two constructions are based on the idea of constructing new codes from old, which produces several new classes of optimal H-LRCs whose lengths can reach up to$q^{2}+q$or unbounded. The final construction generates a class of new optimal cyclic H-LRCs whose lengths divide$q-1$. Compared with the previously known ones, our constructions are new in the sense that their parameters are not covered by the codes available in the literature.

Insertion–deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any$[n,k]$linear code over$\mathbb {F}_{q}$satisfies$d\leq 2n-2k+2$. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Ducet al.showed that the minimum insdel distance of any$[n,k]$Reed-Solomon code is no more than$2n-2k$if$q$is large enough compared to the code length$n$; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any$[n,k]$linear code over$\mathbb {F}_{q}$satisfies$d\leq 2n-2k$if$n > k > 1$. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a 2-D Reed-Solomon code of length$n$over$\mathbb {F}_{q}$is exactly equal to$2n-4$. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal 2-D Reed-Somolom codes meeting the bound.

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let$n>3$be an odd prime number and$q=2^{n}$; let$r\geq 3$be a positive integer satisfying$\gcd (r,n)=1$and$\gcd \big (r,q(q^{2}-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length$q+1$and degree$r$.

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.

For any given positive integer m, a necessary and sufficient condition for the existence of Type-I m-adic constacyclic codes is given. Furthermore, for any given integer s, a necessary and sufficient condition for s to be a multiplier of a Type-I polyadic constacyclic code is given. As an application, some optimal codes from Type-I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant maximum distance separable codes, are constructed.

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.