BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes withd= δ, wheredis the minimum distance of aq-ary BCH code of lengthn, designed distance δ, and offsetb, denoted by C(q,n,δ,b). For primitive BCH codes, we obtain infinite families of BCH codes over F3and F4satisfyingd= δ, where δ 2 {5, 6, 7, 8}. Moreover, we construct several infinite families ofq-ary BCH codes withd= δ, where 2 ≤ δ ≤q−1. For δ =qt+1, we prove that the BCH code C(q,qm−1,qt+1,1)hasd= δ for allmsatisfyingm≡ 0 (modpt), wherepdenotes the characteristic of Fq. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of C(q,qm−1,qt+1,1)is always equal to its Bose distancedB. Our result confirms this conjecture for the casem≡ 0 (modpt). For non-primitive BCH codes, we construct a family of BCH codes C(q, qp−1/λ ,p+1,1)withd= δ =p+ 1, wherepis an odd prime,q = pewithp∤eand λ |q− 1.

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary andp-ary cyclic codes for odd primepin the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over Fqfor each evenq≥ 4. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters [2n,k,d≥ (n− 1)/ log2n], wheren= 2m− 1 andk≥n, are constructed. In addition, 26 repeated-root cyclic codes of length up to 254 over Fqforqϵ {2, 4, 8} with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

MDS codes with determined Hermitian hull dimensions have attracted significant attention for their application in quantum error correction. From an MDS code over Fq2with fixed Hermitian hull dimension ℓ, whereqis a prime power larger than 2, one can obtain an MDS code with any smaller ℓ′-dimensional Hermitian hull for 0 ≤ ℓ′ ≤ ℓ. Then it is natural to consider the problem of determining the maximum Hermitian hull dimension, denoted byLq(n, k), among all MDS codes with the same lengthnand dimensionkover Fq2. Some constructions of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes had been proposed, which addressed this problem for certain parameter regimes. However, it is still unknown for many cases, in particular fork≥q+ 1. In this paper, we study the Hermitian hulls of a class of codes which generalizes GRS codes, called twisted generalized Reed-Solomon (TGRS) codes. TGRS codes contain MDS subclasses that are not linearly equivalent to GRS codes (called non-GRS codes). We give a bound on the Hermitian hull dimensions of certain TGRS codes of general twists. In addition, we derive a lower bound onLq(n, k) forn|q2− 1 and 1 ≤k≤n, which generalizes and improves some previous results. For some parameter regimes wheren≥q+ 1 andk≥q+ 1, we prove thatLq(n, k) ≥k/2 and explicitly construct [n, k]q2MDS codes whose Hermitian hulls have dimension at leastk/2. This result solves partially an open problem pointed out in the literature. The constructed MDS codes arise from either GRS or non-GRS TGRS codes. Furthermore, some sufficient conditions for TGRS codes with general twists to be Hermitian self-orthogonal are given, and Hermitian self-orthogonal non-GRS MDS codes are constructed. Based on our constructions, we provide several families of MDS entanglement-assisted quantum error-correcting codes.

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vlăduţ-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vlăduţ-Zink-like bound and the Gilbert-Varshamov-like bound.

A linear code with parameters [n, k, n–k+ 1] is called maximum distance separable (MDS), and one with parameters [n; k; n–k] is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorialt-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions fort> 2, all with length at mostq+ 1. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater thanq+ 1 are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the firstgeneric constructionofq-ary NMDS codes supporting 2-designs with lengthsexceedingq+1. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

In this paper, we present an antiGriesmer lower bound on the subcode support weights of projective linear codes. This bound is a lower bound on the maximumr-dimensional subcode support weights of projective linear codes. Based on the r-dimensional subcode support weight distributions (r-SSWDs) of linear codes, we compute ther-SSWDs for their simplex complementary codes. Then we construct several infinite families of distance-optimal codes meeting the ℓ-generalized Hamming weight Griesmer bound. These codes do not achieve the Griesmer bound for thej-generalized Hamming weight, where 1 ≤j< ℓ. Moreover, subcode support weight distributions of these optimal linear codes are determined.

Symbol-pair codes were proposed for the application in high density storage systems, where it is not possible to read individual symbols. Yaakobi, Bruck and Siegel proved that the minimum pair-distance$d_{2}$of binary linear cyclic codes satisfies$d_{2} \geq \lceil 3d_{H}/2 \rceil $and introduced b-symbol metric codes in 2016. In this paper, covering codes in b-symbol metrics are considered. Some examples are given to show that the Delsarte bound and the Norse bound for covering codes in the Hamming metric do not hold true for covering codes in the pair metric. We give the redundancy bound on covering radius of linear codes in the b-symbol metric and give some optimal codes attaining this bound. Then we prove that there is no perfect linear symbol-pair code with the minimum pair-distance 7 and there is no perfect b-symbol metric code if$b\geq \frac {n+4}{2}$. Moreover a lot of cyclic and algebraic-geometric codes are proved non-perfect in the b-symbol metric. The covering radius of the Reed-Solomon code in the b-symbol metric is determined. As an application, the generalized Singleton bound on the sizes of list-decodable b-symbol metric codes is also presented. Then an upper bound on lengths of general MDS symbol-pair codes is proved.

In their fundamental paper published in 1965, G. Solomon and J. J. Stiffler invented infinite families of codes meeting the Griesmer bound. These codes are then called Solomon-Stiffler codes and have motivated various constructions of codes meeting or close the Griesmer bound. However weight distributions of Solomon-Stiffler codes have been only determined for very special cases. In this paper, we give a geometric construction of affine and modified affine Solomon-Stiffler codes. Projective Solomon-Stiffler codes are special cases of our modified affine Solomon-Stiffler codes. Several infinite families ofq-ary Griesmer, optimal, almost optimal, two-weight, three-weight, four-weight and five-weight linear codes are constructed as special cases of our construction. Weight distributions of these Griesmer, optimal or almost optimal codes are determined explicitly. Many optimal linear codes documented in Grassl’s list are re-constructed as (modified) affine Solomon-Stiffler codes. Several infinite families of optimal or Griesmer codes were constructed in Shi et, al., IEEE Trans. Inf. Theory, vol. 63, no. 10, 2017, and in Liu et, al., IEEE Trans. Inf. Theory, vol. 65, no. 5, 2019, via Gray images of codes over finite rings. Parameters and weight distributions of these Griesmer or optimal codes can be realized as very special cases in our construction. We also indicate that more general optimal binary linear codes than that constructed in Mondal, IEEE Trans. Inf. Theory, vol. 70, no. 7, 2024, can be obtained from subcodes of codimension one in the binary Solomon-Stiffler codes.

Sum-rank-metric codes have wide applications in multishot network coding and distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp decoding algorithms have been proposed and studied. In this paper, we construct binary linear sum-rank-metric codes in F2×22⊕ F2×22⊕ . . . ⊕ F2×22from BCH, Goppa and additive quaternary codes. A reduction of decoding of binary sum-rank-metric codes to decoding of Hamming metric codes is given. Fast decoding algorithms of BCH-type and Goppa-type binary linear sum-rank-metric codes in F2×22⊕ F2×22⊕ . . . ⊕ F2×22with the block lengthl, which are better than these sum-rank BCH codes, are presented. These fast decoding algorithms for BCH-type and Goppa-type binary linear sum-rank-metric codes need at mostO(l2) operations in the field F4. Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes with the matrix size 2×2 are constructed from Goppa codes.

Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths$n_{i}$and minimum distances$d_{i} \geq \frac {1}{2} \sqrt {n_{i}+2}$was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields${\mathrm { F}}_{2^{s}}$with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields${\mathrm { F}}_{2^{s}}$with a lower bound better than the square-root bound for$s \geq 2$. Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields${\mathrm { F}}_{2^{2s}}$with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over${\mathrm { F}}_{q}$with$q \equiv 1 \pmod {4}$with a square-root-like lower bound are also constructed in this paper.

As cyclic codes and maximum distance separable (MDS) codes, cyclic MDS codes have very nice structures and properties, which have been intensively investigated in literature due to their theoretical interest and practical importance. Particularly, abundant cyclic MDS codes have been determined and constructed for many parameters and most of them were proved to be equivalent to generalized Reed-Solomon (GRS) codes. Hence it is a challenging task to construct non-Reed-Solomon type cyclic MDS codes. In this work, we obtain many new cyclic MDS codes for certain parameters by determining the solutions of the system of polynomial equations. Moreover, by determining the dimension of the Schur square of an MDS code, we can easily show that all of our constructed codes are not equivalent to GRS codes.

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.

Based on known bounds for relative generalized Hamming weights of linear codes, we provide several new bounds for generalized column distances of convolutional codes, including the Griesmer-type bound for generalized column distances. Then we construct several infinite families of convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is small compared with the length of these convolutional codes by using cyclic codes, negacyclic codes and GRS codes. In particular, we obtain some convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is zero or one. Next we prove that the 2-generalized column distance sequence$\{d_{2,j}(\mathcal {C})\}_{j=1}^{\infty }$of any convolutional code$\mathcal {C}$is increasing and bounded from above, and the limit of the sequence$\{d_{2,j}(\mathcal {C})\}_{j=1}^{\infty }$is related to the 2-generalized Hamming weight of the convolutional code$\mathcal {C}$. For$i\ge 3$, we prove that thei-generalized column distance sequence$\{d_{i,j}(\mathcal {C})\}_{j=\lceil \frac {i}{k}-1\rceil }^{\infty }$of any convolutional code$\mathcal {C}$is bounded above and below.

Self-orthogonal and dual-containing codes are two important subclasses of linear codes in coding theory and have been studied for many years. In this paper, we present several sufficient conditions for self-orthogonal or dual-containing codes when a linear code, cyclic code or BCH codeCis transformed to an equivalent code v ·C. Specifically, we prove that linear codes are equivalent to Euclidean or Hermitian self-orthogonal codes if the dimension is very small. For primitive BCH codes, we prove that when designed distances are small, equivalent Euclidean dual-containing codes always exist. From our method presented in this paper, many self-orthogonal or dual-containing linear, cyclic or BCH codes with good parameters can be constructed explicitly. We also construct some Euclidean dual-containing binary BCH codes with best-known parameters.

In eight published papers in IEEE Transactions on Information Theory, infinite families of optimal few-weight binary andq-ary linear codes were constructed and their weight distributions were determined. These codes are linear codes meeting the Griesmer bound. We indicate that many Griesmer codes constructed in these papers are not new. They are actually Solomon-Stiffler codes invented in 1965. Therefore weight distributions of some special binary orq-ary Solomon-Stiffler codes were determined in the papers mentioned above. From a similar geometric approach as Solomon-Stiffler codes, we construct ten infinite families of binary, ternary and quaternary few-weight, optimal, almost optimal and near-optimal linear codes close to the Griesmer bound and their weight distributions are determined. These linear codes have positive Griesmer defects up to five, and thus not Solomon-Stiffler codes and Griesmer codes from minihypers. Moreover, many optimal, best known and almost optimal quantum codes of small lengths, comparing with Grassl's table on quantum codes, are constructed from the same geometric approach as binary Solomon-Stiffler codes.

Sum-rank-metric codes have attracted lots of attention due to their numerous applications, including muti-shot linear network coding, space-time coding, and distributed storage systems. In this paper, we focus on constructing linear sum-rank-metric codes achieving Singleton bound, which are called maximum sum-rank distance (MSRD) codes. This family of codes is the analogue of maximum distance separable (MDS) codes in Hamming metric. We propose two constructions of linear MSRD codes with various matrix sizes. Each of them yields new MSRD codes with different parameter regimes, and one of them generalizes some recent results of Byrne et al. (2021) and Chen (2023). The block lengths and the matrix block sizes of our codes are not restricted to the sizes of the finite field. Our technique is mainly based on rank metric codes and their sub-codes of different minimum rank distances.

Existing reversible audio watermarking (RAW) techniques are often vulnerable to intentional or even unintentional attacks on the cover object. This paper proposes a robust RAW scheme based on lattices, which is referred to as Meet-in-the-Middle Embedding (MME). In MME, the lattice quantization errors are properly scaled and added back to the quantized host signals such that the receiver can estimate the cover. Scaling factor serves as a key factor to the reversibility of MME, whose feasible range is rigorously justified. Both theoretically and experimentally, we demonstrate the superiority of MME to improved quantization index modulation (IQIM) in terms of signal-to-watermark ratio (SWR) and generalized signal-to-noise ratio (GSNR). Moreover, simulations show that MME also outperforms other state-of-the-arts in SWR, objective difference grade (ODG), and bit error rate (BER).

This paper delves into the challenges of spread spectrum (SS) watermarking extraction, considering both reference-free and referential extraction scenarios, within the framework of lattice decoding. The orthogonality of carriers plays a crucial role in the accuracy of extraction, impacting the bit error rate (BER). When carriers lack sufficient orthogonality, conventional reference-free extraction methods such as multi-carrier iterative generalized least-squares (M-IGLS) and referential extraction techniques like MMSE-based schemes encounter performance degradation, posing difficulties in accurately recovering hidden data at the receiver end. To address these challenges, we propose two novel SS watermarking extraction approaches by integrating precise lattice decoding algorithms. Firstly, we introduce the highly accurate yet computationally efficient successive interference cancellation (SIC) algorithm to augment M-IGLS, resulting in a new method termed multi-carrier iterative successive interference cancellation (M-ISIC). Secondly, we adapt the near-optimal sphere decoding (SD) technique for referential extraction in SS watermarking. Theoretical analysis and experimental simulations showcase that our proposed M-ISIC and SD methods outperform M-IGLS and MMSE-based detectors, particularly in scenarios where carrier orthogonality is limited, achieving lower BER. Our code is available athttps://github.com/shx-lyu/M_ISIC.

It is always interesting and important to construct non-Reed-Solomon type MDS codes in coding theory and finite geometries. In this paper, we prove that many non-Reed-Solomon type MDS codes from arbitrary genus algebraic curves can be constructed. It is proved that MDS algebraic geometry (AG) codes from higher genus curves are not equivalent to MDS AG codes from lower genus curves. For genus one case, we construct MDS AG codes of small consecutive lengths from elliptic curves. New self-dual MDS AG codes over F2sfrom elliptic curves are also constructed. These MDS AG codes are not equivalent to Reed-Solomon codes, not equivalent to known MDS twisted Reed-Solomon codes and not equivalent to Roth-Lempel MDS codes. Hence many non-Reed-Solomon type MDS AG codes, which are not equivalent to known MDS twisted-Reed-Solomon codes and Roth-Lempel MDS codes, can be obtained from arbitrary genus algebraic curves. It is interesting open problem to construct explicit longer MDS AG codes from maximal curves.

In this paper, we give many new Singleton type upper bounds on the sizes of codes with given minimum Hamming distances. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes are presented. Our generalized Singleton type upper bounds have wide applications to symbol-pair codes, insertion-deletion codes and locally recoverable codes. The generalized Singleton type upper bounds on symbol-pair codes and insertion-deletion codes are much stronger than the direct Singleton bounds on symbol-pair codes and insertion-deletion codes when the lengths are large and the Hamming minimum distances are small. Upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal$(r, \delta)$locally recoverable codes with any given minimum distance are also presented.

Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of q-ary negacyclic codes of length$(q^{m}-1)/2$and an infinite class of q-ary constacyclic codes of length$(q^{m}-1)/(q-1)$are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.

In this paper, we construct infinitely many families of distance-optimal binary BCH codes with the minimum distance 6 and an infinite family of distance-optimal quaternary BCH codes with the minimum distance 4. We also construct several infinite families of cyclic and negacyclic BCH codes over${\mathbf { F}}_{2}$,${\mathbf { F}}_{3}$,${\mathbf { F}}_{4}$,${\mathbf { F}}_{5}$,${\mathbf { F}}_{7}$and${\mathbf { F}}_{9}$with good parameters$n,\,k,\,d$, such that the maximal possible minimum distance$d_{\max }$of a linear$[n, k]_{q}$code is at most$d_{\max } \leq d+8$. Many codes in these families have optimal or best known minimum distances. 145 optimal or best known codes are constructed as cyclic codes, negacyclic codes, their shortening codes and punctured codes. Several infinite families of rate$\frac {1}{2}$negacyclic$\left [{{n, \frac {n+1}{2}, d}}\right]_{q}$codes or$\left [{{n, \frac {n}{2}, d}}\right]_{q}$codes, such that their minimum distances satisfy$d\geq \frac {cn}{\log _{q} n}$, where c is a positive constant, are also constructed. These are first several families of such negacyclic codes reported in the literature.

In this paper, we give a new upper bound on sizes of linear codes related to weight distributions of codes as follows. Let C be a linear$[n,k,d]_{q}$code, such that, between d and$d\left ({{1+\frac {1}{q-1}}}\right)-1$, the largest weight of codewords in C is the weight$d\left ({{1+\frac {1}{q-1}}}\right)-1-v$, then$k \leq n-d\left ({{1+\frac {1}{q-1}}}\right)+2+v$. Some infinite families of linear codes with arbitrary minimum distances attaining this bound are constructed. This bound is stronger than the Singleton bound for linear codes. Hence we prove that there is no codeword of weights in the range$\left [{{\frac {qd}{q-1}-v,\frac {qd}{q-1}-1}}\right]$for a linear$[n,k,d]_{q}$code, if$v=\frac {qd}{q-1}+k-n-2 \geq 2$. This is the first such kind of result, which concludes vanishing partial weight distributions from four parameters$n,k,d$and q. Then we give vanishing partial weight distribution results for many best known linear codes, some almost MDS codes, general small Griesmer defect codes, some BCH codes, and some cyclic codes. Upper bounds on the number of nonzero weights of binary Griesmer codes and some small Singleton defect codes are also given.

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorialt-designs. It is well-known thatq-ary self-dual cyclic codes exist only whenqis an even prime power andq-ary self-dual negacyclic codes exist for any odd prime powerq. In 2009 a family of binary self-dual cyclic codes with lengthsniand minimum distancesdi≥ 1/2√ni, wherenigoes to the infinity ifigoes to the infinity, was constructed. In this paper, we construct several families ofq-ary self-dual negacyclic codes of lengthsnwith their minimum distances larger than or equal ton1/2for various lengthsnand any given odd prime powerq. Whenq∈ {3, 5} and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.

In this paper, we construct many infinite families of distance-optimal codes with new parameters, some of which are BCH codes and quasi-cyclic codes. In particular, we report the first infinite family of binary distance-optimal BCH codes with the minimum distance 8. Secondly, several infinite families of binary BCH codes and quasi-cyclic codes are presented. Many codes in these families have optimal or best known parameters. Thirdly, we construct infinite families of binary cyclic$\left [{{n, \geq \frac {n+1}{2},d}}\right]_{2}$codes with minimum distances$d \geq \lceil \frac {n-1}{\prod _{i=1}^{s}p_{i}}\rceil $,$n=(2^{p_{1}}-1)(2^{p_{2}}-1) \cdots (2^{p_{s}}-1)$,$p_{1}, \ldots, p_{s}$are different primes. Our construction extends the main result of a recent paper published by Sun et al. to much more general binary cyclic codes with various lengths. We also construct an infinite family of binary quasi-cyclic codes with the rate around$\frac {1}{2}$and relative minimum distance lower bounded by$O\left ({{\frac {1}{\log _{2} \log _{2} n}}}\right)$.

The intersection${\mathbf{C}}\bigcap {\mathbf{C}}^{\perp }$(${\mathbf{C}}\bigcap {\mathbf{C}}^{\perp _{h}}$) of a linear code${\mathbf{C}}$and its Euclidean dual${\mathbf{C}}^{\perp }$(Hermitian dual${\mathbf{C}}^{\perp _{h}}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code${\mathbf{C}}$is transformed to an equivalent code${\mathbf{v}} \cdot {\mathbf{C}}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer$h$satisfying$0 \leq h \leq n-1$, a linear$[2n], [n]_{q}$self-dual code is equivalent to a linear$h$-dimension hull code. On the opposite direction we prove that a linear LCD code over${\mathbf{F}}_{2^{s}}$satisfying$d\geq 2$and$d^{\perp } \geq 2$is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over${\mathbf{F}}_{3}$are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.

Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we give three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous larger linear sum-rank-metric codes with the same minimum sum-rank distances as the previous constructed codes can be derived from our constructions. For example several better linear sum-rank-metric codes over${\mathbf{F}}_{q}$with small block sizes and the matrix size$2 \times 2$are constructed for$q=2, 3, 4$by applying our construction to the presently known best linear codes. Asymptotically our constructed sum-rank-metric codes are close to the Gilbert-Varshamov-like bound on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field${\mathbf{F}}_{q}$with various square matrix sizes$n_{1}, n_{2}, \ldots, n_{t}$satisfying$n_{i} \geq n_{i+1}^{2}+\cdots +n_{t}^{2}$,$i=1, 2, \ldots, t-1$, for any given minimum sum-rank distance. There is no restriction on the block lengths$t$and parameters$N=n_{1}+\cdots +n_{t}$of these linear MSRD codes from the sizes of the fields${\mathbf{F}}_{q}$.

Let${\mathcal C}$be an$[n, k]$linear code over the finite field${\mathbb F}_{q}$. Let$d_{I}({\mathcal C})$denote its insertion-deletion (insdel for short) distance, which characterizes the insdel error-correcting capability of${\mathcal C}$. To determine the insdel distances of linear codes is a very challenging problem. In this paper we propose a strict half-Singleton upper bound$d_{I}({\mathcal C}) \leq 2(n-2k+1)$if${\mathcal C}$does not contain the codeword with all 1s, which generalizes the half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li, and a stronger direct upper bound$d_{I}({\mathcal C}) \leq 2(d_{H}({\mathcal C})-t)$under a weak condition, where$t\geq 1$is a positive integer determined by the generator matrix and$d_{H}({\mathcal C})$denotes the Hamming distance of${\mathcal C}$. A sufficient condition for a linear code attaining the strict half-Singleton bound is given. We prove that the code length of an optimal binary linear insdel code with respect to the (strict) half-Singleton bound is about twice its dimension and conjecture that optimal binary linear insdel codes have exact parameters$[{2k, k, 4}]$or$[{2k+1, k, 4}]$with respect to the half-Singleton bound or the strict half-Singleton bound, respectively. Moreover, interestingly explicit optimal linear insdel codes attaining the (strict) half-Singleton bound, with the code length being independent of the finite field size, are given.

One of the main problems of subspace coding is to determine the maximal size of a constant dimension subspace code with given parameters. In this paper, we show that mixed dimension subspace codes can be used to construct large constant dimension subspace codes. We introduce a new class of subspace codes called mixed dimension/distance subspace codes. Using such codes, we present two constructions for large constant dimension subspace codes. The problem about the sizes of our constant dimension subspace codes is transformed into finding mixed dimension/distance subspace codes with large dimension distributions. The new constructed codes are the largest known for many sets of parameters. Our method gives at least 136 new lower bounds on the sizes of constant dimension subspace codes.

This paper investigates low-dimensional quantizers from the perspective of complex lattices. We adopt Eisenstein integers and Gaussian integers to define checkerboard lattices$\mathcal {E}_{m}$and$\mathcal {G}_{m}$. By explicitly linking their lattice bases to various forms of$\mathcal {E}_{m}$and$\mathcal {G}_{m}$cosets, we discover the$\mathcal {E}_{m,2}^{+}$lattices, based on which we report the best known lattice quantizers in dimensions 14, 15, 18, 19, 22 and 23. Fast quantization algorithms of the generalized checkerboard lattices are proposed to enable evaluating the normalized second moment (NSM) through Monte Carlo integration.

In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes. Our bounds are stronger than some previous known bounds. We apply these upper bounds to AGFC codes from some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. A strong upper bound on the insdel distances of Reed-Muller codes with the special coordinate ordering is also given.

For a linear locally recoverable (LRC) code with length n, dimension k and locality r, its minimum distance d satisfies d ≤ n-k+2-⌈k/r⌉. A code attaining this bound is called optimal. Many families of optimal locally recoverable codes have been constructed by using different techniques in finite fields or algebraic curves. However only optimal LRC codes with lengths n >> q and minimum distances restricted to few constants smaller than 9, have been given in previous constructions. No optimal LRC code over a general finite field Fqwith the length n ~ q2and the minimum distance d ≥ 9 has been constructed. In this article we present a general construction of optimal LRC codes over arbitrary finite fields. Over any given finite field Fq, for any given r ∈ {1,2,...,q-1} and given d satisfying 3 ≤ d ≤ min{r+1,q+1-r}, we construct explicitly an optimal LRC code with length n=q(r+1), locality r and minimum distance d. We also give an asymptotic bound of q-ary LRC codes with locality r (r ≤ q-1), which is better than some known previous asymptotic bounds in some parameter range. Moreover many long LRC codes with locality r (r ≤ q-1) and small defect s=n-k+2-⌈k/r⌉-d are also constructed from algebraic curves with many rational points.

A basic problem for the constant dimension subspace coding is to determine the maximal possible size Aq(n, d, k) of a set of k-dimensional subspaces in Fnqsuch that the subspace distance satisfies d(U, V ) = 2k - 2 dim (U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximum rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions, subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem about the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for Aq(n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

The main problem of constant-dimension subspace coding is to determine the maximal possible size Aq(n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V) ≥ d for any two different subspaces U and V in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of L1(L2(x)) - x where L1and L2are q-polynomials over the extension field Fqn. New lower bounds for Aq(4k, 2k, 2k), Aq(4k t 2, 2k, 2k t 1), and Aq(4k t 2, 2(k - 1), 2k t 1) are presented. Many new constantdimension subspace codes better than previously best known codes with small parameters are constructed.