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.

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.

In coding theory, the well-known class of block codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), form a class of cyclic error-correcting codes constructed using polynomials over a finite field. They are used for various critical practical applications in communication and storage due to their efficient encoding and decoding algorithms. In the past sixty years, significant progress has been made in understanding BCH codes’ dimensions and minimum distances. However, there has been limited research on the minimum distances of the dual codes of BCH codes, making it challenging to determine their actual minimum distances. Therefore, developing accurate lower bounds on the minimum distances of the dual codes of BCH codes is crucial and exciting. In this paper, we primarily use the multiplier technique proposed by Huffman and Pless to investigate the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $of the primitive narrow-sense BCH codes with designed distance$\delta $. When$q = p^{e}$with$e \ge 2$, we improve the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $in the ranges$p^{ei}-p^{e-1}+2 \le \delta \le p^{ei+e-1}-p^{e-1}+1$, where$m \ge 2$and$1 \le i \le m-1$. These new lower bounds are much tighter than the previously known bounds in the literature. This technique also applies to the study of binary dual codes$\mathcal {C}_{(2,2^{m}-1,\delta)}^{\perp } $, for which we obtain tight lower bounds for$\delta = 2^{t}$, where$m \ge 5$is odd and$2 \le t \le m-3$is even.

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.

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 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)$.