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.

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.

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 .

In this paper, the sufficient and necessary conditions for the minimum distance of the BCH code$\mathcal {C}_{(q,q+1,3,h)}$to be 3 and 4 are provided, respectively. Let d be the minimum distance of the BCH code$\mathcal {C}_{(q,q+1,3,h)}$. The following results are proved: 1) for any q,$d=3$if and only if$\gcd (2h+1,q+1)\gt 1$; 2) for q odd,$d=4$if and only if$\gcd (2h+1,q+1)=1$. By combining these conditions with the dimensions of these codes, the parameters of this BCH code are determined completely when q is odd. Moreover, several infinite families of almost maximum distance separable (almost MDS or AMDS for short) codes are derived. Furthermore, a sufficient condition for these almost MDS codes to be distance-optimal and dimension-optimal locally repairable codes is presented. Based on these conditions, several examples are also given.

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.

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.

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field$\mathbb {F}_{2^{n}}$of order$2^{n}$. We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct$t$-designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of$t$-designs for$t \geq 2$. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over${\mathrm{ GF}}(p^{m})$holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over${\mathrm{ GF}}(p^{m})$holding an infinite family of 2-designs for any field${\mathrm{ GF}}(q)$. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over${\mathrm{ GF}}(2^{m})$holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible.

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.

Motivated by the recent studies on Z2Z4-additive cyclic codes, Z2Z2[u]-cyclic codes and Z2Z2(s)-additive codes have been introduced by Aydogdu et al.. In this paper, we study ZqR-linear cyclic codes where R = Zq+ uZq, q is a prime number and u2= 0. There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring ZqR, the spanning sets, the types and sizes of ZqR-cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of q-ary cyclic codes with optimal parameters derived from the ZqR-cyclic codes.

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.

In this study, first the authors present a simplified representation of the Peng–Fan bounds on the periodic Hamming correlation of frequency‐hopping sequence (FHS) sets, which may also be used to check the optimality of an FHS set with respect to the Peng–Fan bounds. Second, they propose a recursive construction of FHS sets from the known ones using some injective functions and the Chinese remainder theorem. It generalises the previous construction of optimal FHSs and FHS sets with composite lengths employing a given function. Without the limit of the specific function, their construction can produce new optimal FHSs and FHS sets that cannot be produced by the earlier construction. By choosing appropriate injective functions and known optimal FHSs and FHS sets, infinitely many new optimal FHSs and FHS sets can be recursively obtained.

In this correspondence, we prove that the distributions of the two-dimensional Kloosterman sums are completely determined by the distributions of the (one-dimensional) Kloosterman sums, and, by a famous result of Lauchad and Wolfman, we can give all the values of the two-dimensional Kloosterman sums. Furthermore, using the identities on one-dimensional Kloosterman sums found by Tor Helleseth and Victor Zinoviev, Hollmann and Xiang, etc., we present some new identities on Kloosterman sums in this note. Meanwhile, we also provide a new approach to producing new Kloosterman sums' identities

We give some new methods for constructing perfect arrays. Some results on the liftability of the perfect ternary sequences are obtained. In addition, we present a new perfect ternary array with index group Z/sub 2//sup r/.