Symbol-pair codes are proposed to guard against pair-errors in symbol-pair channels, where the outputs are overlapping pairs of symbols. It has been a main problem to find symbol-pair codes with relatively large pair distance. In this paper, we leverage cyclic codes over the finite ring 𝔽_q[u]/⟨ u^2⟩ to construct almost MDS symbol-pair codes. A lower bound on the pair distance of the Gray image of a cyclic code over 𝔽_q[u]/⟨ u^2⟩ is derived in terms of the Hamming distances of the torsion code and residue code. Almost MDS symbol-pair codes with pair distance 8 are obtained from the Gray images of cyclic codes over 𝔽_q[u]/⟨ u^2⟩ .

Minimal linear codes are of interest due to their wide application in secret sharing schemes and secure two-party computation. In this paper, we construct several families of binary minimal linear codes from generic construction. In this paper, their weight distributions are determined, and the sufficient conditions for them to be minimal are given. It is worth noting that some minimal binary linear codes obtained in this paper violate the Ashikhmin–Barg condition and can be used to design a secret sharing scheme with a good access structure.

Objective The study of weight distributions of linear codes is fundamental in both theory and applications.Weight distributions indicate the error-correcting capability of a code and allow the calculation of error probabilities for detection and correction.Linear codes with few weights also find applications in secret sharing,strongly regular graphs,association schemes,and authentication codes.Therefore,the construction of linear codes with few weights has attracted sustained attention.Subfield codes of linear codes over finite fields have recently received considerable interest because they can yield optimal codes with potential applications in data storage systems and communication systems.In recent years,subfield codes of linear codes over finite fields with good parameters have been widely studied.Motivated by these constructions,a different defining set is selected to extend existing results.The objectives of this paper are to study the weight distributions and dual codes of this class of linear codes and their punctured codes,and to investigate their subfield codes to obtain linear codes with few weights. Methods The selection of the defining set is a key step in the analysis.The calculation of weight distributions relies on decomposing elements of finite fields into their subfields and applying the first four Pless power moments.Using known results on Kloosterman sums over finite fields,the lengths and weight distributions of this class of linear codes admit closed-form expressions and are completely determined in the binary case.The parameters of their dual codes are also determined and are optimal or almost optimal in the binary case.Trace representations of the subfield codes of this class of codes and their punctured codes are derived.Properties of characters over finite fields are then used to determine the parameters,weight distributions,and dualities of these subfield codes. Results and Discussions By selecting an appropriate defining set and using Kloosterman sums over finite fields,the parameters and weight distributions of a family of q-ary linear codes with few weights and their punctured codes are completely determined.Their dual codes and subfield codes are also examined and are shown to be length-optimal and dimension-optimal with respect to the Sphere-packing bound.A class of eight-weight linear codes and their punctured codes is constructed.The corresponding dual codes are all AMDS linear codes,and they are length-optimal and dimension-optimal linear codes with respect to the Sphere-packing bound(see Theorems 1 and 2,and Tables 1 and 2).The parameters and weight distributions of their subfield codes and the corresponding dual codes are provided(see Theorem 3 and Table 3).In addition,the subfield codes of the punctured codes are studied,and the weight distributions and duality of these codes are determined(see Theorem 4 and Table 4).All results are verified using Magma through two examples. Conclusions A family of q-ary linear codes with few weights and their punctured codes is studied.Based on Kloosterman sums over finite fields,the weight distributions and parameters of the codes and their dual codes are determined,yielding optimal linear codes with respect to the Sphere-packing bound.The weight distributions of their subfield codes and the parameters of the corresponding dual codes are also determined,resulting in few-weight binary linear codes.

For an [n, k, d]q linear code C, the singleton defect of C is defined by S(C) = n-k + 1-d. If S(C) = S(C perpendicular to) = 1, C is called a near maximum distance separable (NMDS) code, where C perpendicular to is the dual of C. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. For a given linear code C of length n over Fq and a nonzero vector u E Fqn, Sun, Ding and Chen [44] defined an extended linear code C(u) of C, which is a generalisation of the classical extended code C(-1) of C. In this paper, for an [n, k]q NMDS code C and a nonzero vector u E Fqn, we provide a sufficient and necessary condition for the extended code C(u) to be an [n + 1, k]q NMDS code. As an application, we construct two classes of NMDS codes of length q + 3 by extending the Roth-Lempel codes of length q + 2. The weight distribution of these two classes of NMDS codes are also given. In addition, more NMDS codes are obtained from the NMDS code C(u), and the covering radius and deep holes of some Roth-Lempel codes are determined. Moreover, we prove that the constructed NMDS codes are optimal locally recoverable codes. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let R be finite non-chain rings 𝔽_q^2m+u𝔽_q^2m , where 𝔽_q^2m is a finite field with q^2m elements, q is an odd prime power, m is a positive integer, u is an indeterminate with u^2=1. In this paper, we firstly define a class of Gray maps, which changes Hermitian dual-containing property of linear codes over R into the Hermitian dual-containing property of linear codes over 𝔽_q^2m . Applying Hermitian construction, a class of q^m -ary quantum codes are obtained from Hermitian constacyclic dual-containing codes over R. Moreover, a family of q-ary primitive quantum BCH codes are determined via the Hermitian construction from the subfield subcodes of Gray images of Hermitian dual-containing u-constacyclic codes over R. Finally, we define another class of maps, which changes the Hermitian dual-containing property of linear codes over R into the trace dual-containing property of linear codes over 𝔽_q^2m . Using Symplectic construction, another class of q^m -ary quantum codes are obtained from Hermitian dual-containing u-constacyclic codes over R.

Self-dual codes have been studied for decades since their important applications in cryptography and combinatorics. In this paper, we construct an infinite family of q2-ary Hermitian self-dual constacyclic codes with length n = q2m-1 & micro; and minimum distance not less than root n, where q is an odd prime power and & micro; is an even factor of q2m-1. We also present an infinite family of q2-ary Hermitian self-dual cyclic codes with length 2n = 2(q2m-1) & micro; and minimum distance not less than root 2n for even prime power q and odd factor & micro; of q2m-1. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Self-orthogonal codes have received much attention in recent years due to their wide-ranging practical applications. The aim of this paper is to construct several infinite families of self-orthogonal codes by using cyclic codes with few nonzeros. We present a sufficient condition for q-ary cyclic codes to be self-orthogonal. Based on this condition, we obtain several families of self-orthogonal cyclic codes over 𝔽_q . These self-orthogonal codes contain many optimal or best-known codes. The dimensions of these codes are determined and the lower bounds on their minimum distances are given. Numerous examples demonstrate that the lower bounds on the minimum distances of the codes we construct are tight.

Let q≥ 3 be an odd prime power and m≥ 2 be an even integer. In this paper, a kind of narrow-sense constacyclic BCH codes over 𝔽_q^2 with lengths n=q^2m-1/2a(q+1) are studied, where q≡ a+1 2a and a

The construction of self-orthogonal codes is an interesting topic due to their wide applications in communication and cryptography. In this paper, we construct several families of self-orthogonal cyclic codes with length n = q(m)-1/lambda , where lambda | q - 1 and m >= 3 is odd. It is proved that there exist q-ary self-orthogonal cyclic codes with parameters [ n, n - 1/2 >= d] for even prime power q , and [ n, n - 1/2, >= d ] or [ n, n - 1/2, >= d ] for odd prime power q , where d is significantly better than the square-root bound. These several families of self-orthogonal cyclic codes contain some optimal linear codes.

Self-orthogonal codes have a wide range of applications in various fields, especially communication and cryptography. In this paper, we construct four families of linear codes over finite fields via a defining-set construction. The weight distributions of these codes are determined. We show that most of these codes are self-orthogonal and reach the Grismer bound. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we investigate negacyclic dually-BCH codes over GF(q) of length n=q^m-1 . We provide the first three largest odd coset leaders modulo 2n . Sufficient and necessary conditions in terms of designed distance for negacyclic codes over GF(q) of length n=q^m-1 to be negacyclic dually-BCH codes are presented.

Linear codes with complementary duals (LCD codes) have attracted a lot of interest in recent years due to their applications in data storage, protections against fault injection attacks and implementations against side-channel attacks. The objective of this paper is to investigate some ternary LCD cyclic codes with length n=3^2m-1/4 . The dimension of LCD BCH codes 𝒞_(3,n,2δ ,-δ +1) is determined, where 2≤δ≤3^m+1 . Moreover, we calculate the minimum distance of some of these LCD BCH codes with specific designed distance. By calculating the first three largest absolute coset leaders modulo n , we construct several classes of ternary LCD cyclic codes. The weight distribution of some of them is given.

In this paper, a class of narrow-sense constacyclic BCH codes over 𝔽_q^2 with length n=q^2m-1/2( q^2-1) is studied, where q≥ 3 is an odd prime power and m≥ 2 is even. The maximum designed distance such that narrow-sense constacyclic BCH codes over 𝔽_q^2 with length n containing their Hermitian dual codes is determined. We obtain some new quantum codes by using such narrow-sense constacyclic BCH codes. Our constructions not only have larger designed distance but also have better parameters than the ones in the literature.

. Symbol-pair codes can be applied to protect against the pair errors in symbol-pair read channels. Constructing symbol-pair codes with good parameters has become a main problem in the study of pair-error correction. In this paper, we give a new class of almost maximum distance separable (AMDS) symbol-pair codes with length 8p through repeated-root cyclic codes, where p equivalent to 1(mod 8). The results are obtained by analyzing the solutions of certain equations over finite fields.

Constacyclic BCH codes are an interesting subclass of constacyclic codes because of their important theoretical and practical value. The purpose of this paper is to study the parameters of cyclic BCH codes of length n = q^m - 1 and negacyclic BCH codes of length n = q^m - 1/2 . We settle completely their dimensions. We also determine the minimum distances of a class of cyclic BCH codes of length n = q^m - 1 and give a lower bound on the minimum distances of other classes of constacyclic BCH codes. As seen by the code examples in this paper, the lower bound on the minimum distances of constacyclic BCH codes we gave is very close to the true minimum distances. These q -ary codes have good parameters in general.

BCH codes have been widely used in consumer devices, communication systems, and data storage systems. In this paper, we study the dimensions of four families of the BCH codes C-(q,C-n,C-delta,C-0) of length n = q(m)+ 1, where designed distances dare given as follows. (1) delta = a q(m)+1/q+ 1+ 1, where 1 <= a <= (sic)q-1/2(sic) and m is odd. (2) delta = a q(m)-1/q-1+2, where 1 <= a <= (sic)q-1/2(sic). (3) a q(m)-1/q-1+ 2 < delta < a q(m)-1/q-1 + (q - 2a - 1)q(m-1/2) + 3, where mis odd and 0 <= a <= (sic)q-3/2(sic). (4) a q(m)-1/q-1+ 2 < delta < a q(m)-1/q-1+ q(m/2) + 4, where mis even and 0 <= a <= (sic)q-3/2(sic). The minimum distance of the first family of the BCH codes C-(q,C-n,C-delta,C-0) is determined. The Bose distances of the third and fourth families of the BCH codes C-(q,C-n,C-delta,C-0) are determined under some conditions. Many optimal linear codes are obtained from these BCH codes. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let n=2(p^m-1)/(p-1) , where p is an odd prime and m>1 is a positive integer. In this paper, we research optimal p-ary constacyclic codes with two zeros. Two classes of optimal p-ary [n,n-2m,4] constacyclic codes are presented by searching the solutions of certain congruence equations over 𝔽_p^m . Four explicit constructions of optimal constacyclic codes with such parameters are provided. The dual codes of a subclass of these constacyclic codes are also investigated.

Let R be the finite chain ring 𝔽_p^2m+u𝔽_p^2m , where 𝔽_p^2m is the finite field with p^2m elements, p is a prime, m is a non-negative integer and u^2=0. In this paper, we firstly define a class of Gray maps, which changes the Hermitian self-orthogonal property of linear codes over 𝔽_2^2m+u𝔽_2^2m into the Hermitian self-orthogonal property of linear codes over 𝔽_2^2m . Applying the Hermitian construction, a new class of 2^m -ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over 𝔽_2^2m+u𝔽_2^2m. We secondly define another class of maps, which changes the Hermitian self-orthogonal property of linear codes over R into the trace self-orthogonal property of linear codes over 𝔽_p^2m . Using the Symplectic construction, a new class of p^m -ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over R.

BCH codes are an especially important kind of cyclic codes, which are extensively used in many domains such as communication and storage systems. Determining the dimensions of BCH codes is a challenging problem in coding theory. In this paper, we study BCH codes of length qm+1 2 , for odd m and m equivalent to 2(mod 4). The first few largest coset leaders of such BCH codes are obtained. On the basis of this, we determine their parameters and obtain a few optimal or almost optimal codes.