In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

We closely study N-sum boxes and their transfer matrices. Recently formulated as an abstraction for linear computations over quantum network, such a box allows for tools from quantum information processing to be applied on classical computational problems. We investigate (k, N + t)-sum boxes, which are generalized version of the N-sum boxes, and propose a general construction of their transfer matrices. Seen in this light, an N-sum box is a special case when k = N and t = 0.

The correlation properties of sequences form a focal point in the design of multiple access systems of communications. A popular choice for the set of sequences to deploy is the quasi-complementary sequence set. There is a growing body of literature that recognises the importance of quasi-complementary sequence sets. In this paper, using additive characters over finite fields, we propose five classes of asymptotically optimal quasi-complementary sequence sets, including periodic and aperiodic ones. In particular, the designed asymptotically optimal quasi-complementary sequence sets have new parameters and small alphabet sizes. The small alphabet size enhances their appeal for implementation.

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed 5. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to 2.

A code is called a punctured concatenated code if it can be obtained by puncturing a concatenated code at suitable coordinates. Based on this new concept, we construct several classes of optimal or almost optimal linear codes. There are two major contributions in this paper. Let the inner code be an [n,m]qlinear code derived from the defining setD= {d1,d2, . . . ,dn}. On the one hand, by employing a maximum distance separable (MDS) code with dimension 2 over Fqmas the outer code, we propose two classes of linear codes with few weights. The duals of these codes are shown to be dimension-optimal with respect to the sphere-packing bound. On the other hand, letq= 2, by choosing an MDS code with dimension 3 over F2mas the outer code, we construct another class of linear codes. The parameters and weight distributions of these codes are completely determined. Furthermore, their dual codes are almost distance-optimal with respect to the sphere-packing bound.

A binary linear code is calledasymptotic frame error rate (AFER)-optimalif it achieves the maximum possible value of the minimum distance while having the smallest value of the corresponding error coefficient. Over the additive white Gaussian noise channel and under maximum-likelihood decoding, AFER-optimal codes attain the best possible asymptotic frame error rate at high signal-to-noise ratio. In this paper, we present several bounds on the smallest error coefficients of binary linear codes and give several constructions of AFER-optimal binary linear codes. Many examples confirm that our bounds are sharp on numerous occasions. In addition, we give two families of AFER-optimal codes that respectively attain the proposed bounds with equality.

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.

Wei’s duality theorem, proposed by Wei in 1991, initially deals with the generalized Hamming weight (GHW) of linear codes over finite fields. Specifically, the Wei’s duality theorem describes the relationship between the GHWs of a linear code and those of its dual code. The GHWs of linear codes are important parameters for measuring the security of codes in certain cryptographic applications. In this paper, we generalize the concept of GHW of linear codes over finite fields to the generalized poset weight (GPW) of linear codes over Galois rings. We also present an extension of GPW, abbreviated as EGPW. Similar to the GHWs, we derive two forms of the Wei’s duality theorem with respect to GPWs and EGPWs. In conclusion, we demonstrate that the GPWs of linear codes serve as indicators for assessing the security of information transmission within the wire-tap channel of type II .

This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction. Subsequently, we propose generalized Construction X, which facilitates the construction of non-integer dimensional optimal additive codes from linear codes. Then, we construct ten classes of optimal quaternary non-integer dimensional additive codes through these two methods. As an application, we also determine the optimal additive$[n,3.5,n-t]_{4}$codes for all t with variable n, except for$t=6,7,12$.

The number of nonzero weights of a linear code is essential in coding theory as it unveils salient properties of the code, such as its covering radius. In this paper, we establish two upper bounds on the number of nonzero weights of a linear code with prescribed automorphism. Our bounds are applicable for almost all linear codes and tighter than previously known bounds. Examples confirm that our bounds are sharp on numerous occasions. In addition, we give an infinite family of linear codes that attain our bounds with equality.

The b-symbol metric is a generalization of the Hamming metric. Linear codes in the b-symbol metric have been used in the read channel whose outputs consist of b consecutive symbols. The Griesmer bound outperforms the Singleton bound for${\mathbb {F}}_{q}$-linear codes in the Hamming metric, when q is fixed and the length is large enough. This scenario is also applicable in the b-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the b-symbol metric. In this paper, we present the b-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the b-symbol Griesmer bound.

Spectral bounds form a powerful tool to estimate the minimum distances of quasi-cyclic codes. They generalize the defining set bounds of cyclic codes to those of quasi-cyclic codes. Based on the eigenvalues of quasi-cyclic codes and the corresponding eigenspaces, we provide an improved spectral bound for quasi-cyclic codes. Numerical results verify that the improved bound outperforms the Jensen bound in almost all cases. Based on the improved bound, we propose a general construction of quasi-cyclic codes with excellent designed minimum distances. For the quasi-cyclic codes produced by this general construction, the improved spectral bound is always sharper than the Jensen bound.

Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.

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).

We consider the repair scheme of Guruswami-Wootters for the Reed-Solomon code and ask: can we correctly repair a failed node in the presence of erroneous nodes? Equivalently, we consider the collection of downloaded traces as a code and investigate its code-distance properties. We propose three lower bounds on its minimum distance and study methods to efficiently correct errors close to these bounds.

Convolutional codes are essential in a wide range of practical applications due to their efficient non-algebraic decoding algorithms. In this paper, we first propose a new family of matrices over finite fields by combining Vandermonde and Moore matrices. Using favourable properties of the matrices in this new family enables us to construct a new family of convolutional codes with memory 1 and maximum distance profile. It is notable that the alphabet sizes of this new family of convolutional codes with maximum distance profile can be kept significantly smaller than those in the literature. Keeping the code rate to a constant, the alphabet size is roughly the square root of the previously best-known value.

Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with$(r,\delta)$-locality that achieve equality in the Singleton-type bound. We use matrix-product codes to propose two constructions of$q$-ary optimal$(r,\delta)$locally repairable codes of lengths up to$q^{2}+q$. The ingredients in the matrix-product codes are linear maximum distance separable codes. We give another construction of optimal$(r,\delta)$locally repairable codes by using optimal locally repairable codes as ingredients in the matrix-product approach. The codes in this third construction have unbounded lengths not divisible by$(r+\delta -1)$. The three constructions of optimal$(r,\delta)$locally repairable codes constructed here are new. Previously constructed codes in the literature have not covered the same sets of parameters. Our construction proposals are flexible since one can easily vary$r$and$\delta $to come up with particular parameters that can suit numerous scenarios.

In emerging storage technologies, the outputs of the channels consist of overlapping pairs of symbols. The errors are no longer individual symbols. Controlling them calls for a different approach. Symbol-pair codes have been proposed as a solution. The error-correcting capability of such a code depends on its minimum pair distance instead of the usual minimum Hamming distance. Longer codes can be conveniently constructed from known shorter ones by a matrix-product approach. The parameters of a matrix-product code can be determined from the parameters of the ingredient codes. We construct a new family of maximum distance separable (MDS) symbol-pair matrix-product codes. Codes which are permutation equivalent to matrix-product codes may have improved minimum pair distances. We present four new families of MDS symbol-pair codes and a new family of almost MDS symbol-pair codes. The codes in these five new families are permutation equivalent to matrix-product codes. Each of our five constructions identifies permutations that can increase the minimum pair distances. We situate the new families among previously known families of MDS symbol-pair codes to highlight the versatility of our matrix-product construction route.

Locally recoverable codes play a crucial role in distributed storage systems. Many studies have only focused on the constructions of optimal locally recoverable codes with regard to the Singleton bound. The aim of this paper is to construct optimal binary locally recoverable codes meeting the alphabet-dependent bound. Using a general framework for linear codes associated to a set, we provide a new approach to constructing binary locally recoverable codes with locality 2. We turn the problem of designing optimal binary locally recoverable codes into constructing a suitable set. Several constructions of optimal binary locally recoverable codes are proposed by this new method. Finally, we propose constructions of optimal binary locally recoverable codes with locality 2 and locality parameters (r,δ) by Griesmer codes.

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound.

By introducing several levels of recoverability for locally recoverable codes (LRCs), LRCs with hierarchical locality (H-LRCs) are defined for correcting different numbers of erasures. There are two major ingredients in this paper. The first is to investigate some properties and existence conditions of optimal H-LRCs with respect to a generalized Singleton-like bound for H-LRCs. The second ingredient is to propose several constructions of optimal H-LRCs by employing cyclic codes. Notably, the parameters of these optimal H-LRCs are flexible.

The hull of linear codes has promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with hulls of arbitrary dimensions are presented. As an application, using these MDS codes with hulls of arbitrary dimensions, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.

In this paper, we propose two constructions of complex codebooks from character sums over Galois rings. The complex codebooks produced by these two constructions are proved to be asymptotically optimal with respect to the Welch bound. In addition, the parameters of the complex codebooks presented in this paper are new.

Codebooks with low-coherence have wide utilization in many fields, such as direct spread code division multiple access communications, compressed sensing and so on. There are two major ingredients in this paper. The first is to present a new character sum, the hyper Eisenstein sum and study the properties of this character sum. As an application, the second ingredient is to propose two constructions of codebooks with the hyper Eisenstein sum. The codebooks generated by these constructions asymptotically meet the Welch bound. The parameters of these codebooks are new.