Self-orthogonal codes are an important type of linear codes since they are very closely related to designs, lattices, and quantum codes. Bose-Chaudhuri-Hocquenghem codes (BCH codes) have various practical applications in communication and storage due to their efficient encoding and decoding algorithms. In this paper, we will focus on the primitive and projective self-orthogonal BCH codes in both Euclidean and Hermitian cases. Our main objective is to characterize primitive and projective Euclidean and Hermitian self-orthogonal BCH codes and investigate their parameters. For the Euclidean case, the primitive and projective self-orthogonal BCH codes are characterized completely by using their designed distances. For the Hermitian case, a sufficient and necessary condition for all primitive BCH codes being self-orthogonal are presented, while the characterizations of projective Hermitian self-orthogonal BCH codes are obtained in some cases. Moreover, the dimensions of some Euclidean and Hermitian self-orthogonal BCH codes are determined explicitly and lower bounds on their minimum distances are given.

The relative hull of a linear codeC1with respect to another linear codeC2is defined as the intersection ofC1and the dual ofC2, i.e.,C1∩C⊥2. Andersonet al. [2] demonstrated that the dimension of the relative hull can be adjusted, either repeatedly increased or decreased by one, until a certain bound is reached by substituting eitherC1orC2with its equivalent code. As a special class of linear codes, BCH codes are both theoretically significant and practically valuable for communication and storage systems due to their good algebraic structures and flexible error-correcting capabilities. This raises an important question: how does the dimension of the relative hull change when C1 and C2 are BCH codes or their equivalent codes? This paper focuses on the relative hull dimensions of narrow-sense primitive BCH codes, building on the insights from [2]. We present several sufficient and necessary conditions based on the designed distances of BCH codes, which ensure that the dimensions of the relative hulls reach the lower or upper bounds for linear codes. Additionally, we study the parameters of the relative hulls of various classes of BCH codes, providing details on their dimensions and developing lower bounds for their minimum distances. Furthermore, we investigate how the dimensions of the relative hulls are affected when the BCH codesC2are replaced by their equivalent codesC′2.

Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let Fpmbe the finite field withpmelements for a primepand a positive integerm. LetV(p)nbe ann-dimensional vector space over Fpfor a primep. In this paper, we study the hybrid character sums of the form Σx∈V(p)nψ (F(x)) χa(x), whereFis a function fromV(p)nto Fpm,a∈V(p)n, ψ is a nontrivial multiplicative character of Fpm, χa(x) = ζ⟨a,x⟩npis the character ofV(p)nand ⟨, ⟩ndenotes a (non-degenerate) inner product ofV(p)n. IfF(x) is a vectorial dual-bent function anda∈V(p)n\ {0}, we determine their complex modulus or explicit values under certain conditions. This generalizes some known results as special cases. We show that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The codebooks we construct have very small alphabet sizes. This enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.

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.

Abstract. Affine-invariant codes are an important class of linear codes, which are extended cyclic codes of length [Formula: see text] invariant under the affine groups acting on [Formula: see text]. These codes are closely related to combinatorics, as they can be applied to construct [Formula: see text]-designs. It is known that the classical Reed–Muller codes and extended primitive narrow-sense Bose–Chaudhuri–Hocquenghem codes are affine-invariant. The objective of this paper is to construct a class of affine-invariant codes [Formula: see text] and investigate the parameters of these codes and their related codes. The dimensions of the codes [Formula: see text] and [Formula: see text] with [Formula: see text] are presented and a recursive formula to compute the dimensions of [Formula: see text] and [Formula: see text] is developed for general [Formula: see text], where [Formula: see text] is the extended code of [Formula: see text]. Meanwhile, lower bounds on minimum distances of [Formula: see text] and [Formula: see text] are also given. Moreover, the parameters and the borders of the dual codes [Formula: see text] are investigated. Two necessary and sufficient conditions for [Formula: see text] being self-orthogonal with [Formula: see text] are developed by employing their borders. In addition, for [Formula: see text] we explore the parameters of the hull of [Formula: see text] and determine its border. It should be pointed out that several affine-invariant self-orthogonal codes will be obtained.

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic (p2n,pn− 1,pn− 1,pn+ 1) QCSSs with small alphabet sizep, which has larger set size than the known family of periodic (pn(pn−1),pn−1,pn−1,pn+1) QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size Θ(K2) for some flock sizeK. Compared with known asymptotically optimal aperiodic QCSSs in the literature, our proposed aperiodic QCSSs have better or new parameters. Particularly, for three families of the costructed aperiodic QCSSs, the column sequence peak-to-average power ratio (PAPR) is upper bounded by p if we select suitable column orthogonal complex matrices.

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.

In this paper, we use extended Euclid’s algorithm to propose new decoding algorithms for two classes of maximum distance separable (MDS) twisted generalized Reed-Solomon (TGRS) codes of parameters$[n, n-t, t+1]$over$\Bbb F_{q}$. For even t, the algorithms can correct$\frac {t}{2}$errors with time complexity$O(qn)$. Moreover, we also give a new decoding algorithm for a class of twisted Goppa codes. For even degree t of a Goppa polynomial, it can also correct$\frac {t}{2}$errors, which generalizes a$\lfloor \frac {t-1}{2}\rfloor $-error-correcting decoding algorithm by Sui and Yue (2023).

The Rosenbloom-Tsfasman metric (RT-metric for short) is a generalization of the Hamming metric. Matrix codes in the frame of the RT-metric have been used in information transmission over parallel channels. In this paper, we develop some new upper bounds on the minimum RT-distance of an [h×n, k, dRT] linear matrix code, which generalize the Singleton-type bound derived by Rosenbloom and Tsfasman. It should be emphasized that the upper bounds build a connection between the RT-metric and the Hamming metric. Constructions of linear matrix codes are presented and their parameters for the RT-metric are investigated. It is shown that every linear matrix code can be expressed by using the trace function, which is a generalization of the well-known defining-set construction of linear codes. Moreover, we obtain several classes of optimal linear matrix codes in this paper.

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.

Abstract Coin mixing is an efficient anonymization technology of cryptocurrency used to eliminate the linkability of transaction parties by hiding their addresses in an anonymous set. However, a common weakness with most existing coin mixing schemes is that the amount of mixed coins must be the same for all requests within a mixing cycle, otherwise it is easy for an attacker to restore the linkability of transaction parties. In this paper, we design a stage-payable puzzle solution mechanism, named CoinFA, which reverses the control of the requesting amounts to the users for flexible mixing amounts. In our design, the payee (with an output address) first requests a puzzle from the mixers and the latter are the only ones who know the solution of the puzzle. If the payee solves the puzzle successfully, he can be rewarded with the corresponding Bitcoins. The payer (allowed to have multiple input addresses) then requests the solution by paying in installments. We achieve better security by weakening the rights of the involved third parties, while the hierarchical structure allows our solution to have better efficiency and robustness. We perform a security analysis on CoinFA based on the standard Rivest-Shamir-Adleman (RSA) assumption and Elliptic Curve Digital Signature Algorithm unforgeability. We also analyze the performance of CoinFA by comparing it with two related schemes, and the results show that our CoinFA scheme has a greater advantage when the mixing amount is relatively small.

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.

The (Schur) squares of linear codes are an interesting research topic in coding theory, and they have important applications in cryptography. Linear complementary dual codes (LCD codes) have been widely applied in data storage, communication systems, consumer electronics, and cryptography. Given these exciting applications of squares and LCD codes, we mainly focus on the squares of LCD cyclic codes in this paper. It will be proved that the square of an LCD cyclic code is still an LCD cyclic code. As a subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have explicit defining sets that include consecutive integers, which gives an advantage of analyzing the parameters of BCH codes and their related codes. We will investigate the squares$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$of the primitive LCD BCH codes$\mathcal {C}(t)$and their complements$\mathcal {C}(t)^{c}$, respectively, where$\mathcal {C}(t)=\mathcal {C}_{(q,q^{m}-1,2t,-t+1)}$is the BCH code of length$q^{m}-1$over$\mathbb{F}_{q}$with designed distance$2t$. Two sufficient and necessary conditions to guarantee that$\mathcal {C}^{2}(t) \ne \Bbb \{\textbf {0}\}$and$\mathcal {C}^{2}(t)^{c} \ne \mathbb{F}_{q}^{n}$are proposed by giving restrictions on designed distances. Furthermore, the dimensions and lower bounds on minimum distances of$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$are presented in some cases. The parameters of the squares of the complements of the Melas codes$M(q,m)$are also investigated.

Binary cyclic codes with parameters$[n,(n+1)/2, d\geq \sqrt {n}]$are very interesting, as their minimum distances have a square-root bound. The binary quadratic residue codes and the punctured binary Reed-Muller codes of order$(m-1)/2$for odd$m$are two infinite families of binary cyclic codes with such parameters. The objective of this paper is to present and analyse an infinite family of binary BCH codes${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1},d]$whose minimum distance$d$much exceeds the square-root bound when$m \geq 11$is a prime. The binary BCH code${\mathcal {C}}(3)$is the binary Hamming code and distance-optimal. The binary BCH code${\mathcal {C}}(5)$has parameters$[{31,16,7}]$and is distance-almost-optimal. The binary BCH code${\mathcal {C}}(7)$has parameters$[{127,64,21}]$and has the best known parameters. In addition, there is no known$[2^{m}-1,2^{m-1}]$binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1}]$for any odd prime$m$.

BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of duals of BCH codes. Recently, a concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in Gong et al., (2022). For a prime power$q$and an integer$m \ge 4$, let$n=\frac {q^{m}-1}{q+1}$($m$even), or$n=\frac {q^{m}-1}{q-1}$($q>2$). In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense BCH codes of length$n$are dually-BCH codes, which extended the results in Gong et al., (2022). Lower bounds on the minimum distances of their dual codes are developed for$n=\frac {q^{m}-1}{q+1}$($m$even). As byproducts, we present the largest coset leader$\delta _{1}$modulo$n$being of two types, which proves a conjecture in Wu et al., (2019) and partially solves an open problem in Li et al., (2017). We also investigate the parameters of narrow-sense BCH codes of length$n$with design distance$\delta _{1}$. The BCH codes presented in this paper have good parameters in general.

Schur product was originally proposed in coding theory for algebraic decoding algorithms and widely applied to solve some cryptographic problems in recent years. This shows the great importance of the Schur product in both coding theory and cryptography. As a well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have wide applications in communication and storage systems. Let$\mathcal {C}_{1}$and$\mathcal {C}_{2}$be two primitive BCH codes over$\Bbb {F}_{q}$with designed distances$\delta _{a}$and$\delta _{b}$, respectively, where$2 \leq \delta _{a}, \, \delta _{b} \leq n$. Let$\mathcal {C}_{1}^{c}$and$\mathcal {C}_{2}^{c}$be the complements of$\mathcal {C}_{1}$and$\mathcal {C}_{2}$, respectively. This paper aims to investigate the parameters of the products$\mathcal {C}_{1} \star \mathcal {C}_{2}$and$\mathcal {C}_{1}^{c} \star \mathcal {C}_{2}^{c}$. We will present some sufficient and necessary conditions to guarantee that$\mathcal {C}_{1} \star \mathcal {C}_{2} \neq \Bbb F_{q}^{n}$and$\mathcal {C}_{1}^{c} \star \mathcal {C}_{2}^{c} \neq \Bbb F_{q}^{n}$by giving restrictions on the designed distances$\delta _{a}$and$\delta _{b}$of the two BCH codes, respectively. The dimensions of these products are determined explicitly and lower bounds on the minimum distance are developed in some cases. Some optimal or best known codes are found. Moreover, it should be emphasized that a class of$[n, k, d]$cyclic codes over$\Bbb F_{q}$with dimension$k \ge \frac {n}{2}$and$d \ge \sqrt {n}$are presented.

Electronic health records (EHRs: digital collections of patient health status and diagnosis) are generally shared, analyzed and stored on cloud servers. One operational challenge is to ensure that EHRs are stored correctly, for example using provable data possession (PDP). Seeking to contribute to the literature, we propose a certificateless PDP scheme for cloud-based EHRs. In our scheme, we distribute multiple copies of EHRs on different cloud servers to allow for corrupted EHRs to be recoverable from other intact copies. The scheme is also designed to resist copy-summation attack which assures that cloud servers are storing EHRs honestly. In our approach, EHRs are stored in ciphertext form so that only authorized users can decrypt and gain access to the information. We also design a new data structure – map version marker table (MVMT) – for block-level dynamic operations and data traceability. Specifically, MVMT allows an authorized doctor to access historical EHRs to inform their diagnosis and decision-making. The security and performance analyses show that our scheme is secure (assuming the intractability of the computational Diffie-Hellman problem) and is practical to support cloud-based EHR applications.

Studying the Schur square of a linear code is an important research topic in coding theory. Schur squares have important applications in cryptography and private information retrieval schemes, notably in secure multiparty computing or designing bilinear multiplication algorithms in finite extensions of finite fields through the notion of supercodes. Thanks to their exciting applications in cryptography, squares and powers of several linear codes have been investigated. In this paper, we will focus on the Schur square of a relevant well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), which have wide applications in communication and storage systems and benefit from explicit defining sets that include consecutive integers, which gives the advantage of analyzing the parameters of BCH codes and their complements. Our main objective is to investigate the parameters of the squares of primitive narrow-sense BCH codes$\mathcal C(\delta)$and their complements$\mathcal C(\delta)^{c}$. We will present two sufficient and necessary conditions to guarantee that$\mathcal C^{2}(\delta) \ne \Bbb F_{q}^{n}$and$\mathcal C^{2}(\delta)^{c} \ne \Bbb F_{q}^{n}$by giving restrictions on designed distance$\delta $, where$2 \le \delta \le n$. Based on these two characterizations, the dimensions and minimum distances of$\mathcal C^{2}(\delta)$and$\mathcal C^{2}(\delta)^{c}$are investigated in some cases. The dimensions of these squares are determined explicitly, and lower bounds on the minimum distance are given.

As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let$\mathcal C$be a$q^{2}$-ary BCH code of length$n$with respect to an$n$-th primitive root of unity$\beta $over an extension field of$\Bbb F_{q^{2}}$, and let$\mathcal C^{\perp H}$denote its Hermitian dual code, where$q$is a prime power. If both$\mathcal {C}$and$\mathcal C^{\perp H}$are a BCH code with respect to an$n$-th primitive root of unity$\beta $, then$\mathcal C$is called aHermitian dually-BCH code. The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length$n$over$\mathbb {F}_{q}$is always relative to an$n$-th primitive root of unity$\beta $in an extension field of$\mathbb {F}_{q}$, and is called a dually-BCH code if its dual is also a BCH code relative to the same$\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances$\delta $will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances$\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection.

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes.

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$be the primitive BCH code over$\mathbb {F}_{r}$of length$r^{m}-1$with designed distance$\delta $, where$\mathbb {F}_{r}$is the finite field of order$r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$and its hull in both Hermitian and Euclidean cases for$2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let C be an [n, k] cyclic code over Fq, where Fqis the finite field of order q. The hull of C is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension k - 1 or k⊥- 1, where k⊥is the dimension of the dual code C⊥. Several sufficient and necessary conditions for primitive and projective BCH codes to have (k - 1)-dimensional (or (k⊥-1)dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained.

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes C(q,m,δ(2))and C(q,m,δ(3))and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes C̅(q,m,δ(2))have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code C⊥(q,m,δ(2))for q ≥ 5. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained.

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let Frbe an extension of a finite field Fqand r = qm, we determine the weight distributions of the cyclic codes C={c(a, b): a, b ∈ Fr}, c(a, b)= Trr/q(ag10+bg20),...,Trr/q(ag1n-1+bg2n-1)), g1, g2∈ Fr, in the following two cases: 1) ord(g1)=n, n|r-1 and g2=1 and 2) ord(g1)=n, g2=g12, ord(g2)=n/2, m=2, and 2(r-1)/n|(q+1).

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let Fr be a finite field with r = qm. Suppose that g1, g2 ∈ F*rare not conjugates over Fq, ord(g1) = n1, ord(g2) = n2, d = gcd(n1, n2), and n = n1n2/d. Let Fq(g1) = Fqm1, Fq(g2) = Fqm2, and Ti denote the trace function from Fqmito Fq for i = 1, 2. We define a cyclic code C(q,m,n1,n2) = {c(a, b) : a ∈ Fqm1, b ∈ Fqm2}, where c(a, b) = (T1(ag01) + T2(bg02), T1(ag11) + T2(bg12), ... , T1(agn-11) + T2(bgn-12)). We mainly use Gauss periods to present the weight distribution of the cyclic code C(q,m,n1,n2). As applications, we determine the weight distribution of cyclic code C(q,m,qm1-1,qm2-1) with gcd(m1, m2) = 1; in particular, it is a three-weight cyclic code if gcd(q -1, m1 -m2) = 1. We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.