Determining the possible codeword weights of a linear code is a central problem in coding theory, as the weight distribution reflects the underlying algebraic structure and governs key properties such as minimality and self-orthogonality. This question becomes particularly challenging when the number of distinct nonzero weights increases. In this work, we study a family of ternary linear codes defined by a set determined by two homogeneous quadratic functions over finite fields. By employing the theory of quadratic forms and their associated exponential sums, we analyze four specific quadratic forms, determine their ranks and signs, and derive explicit formulas for the code lengths and all possible nonzero weights. Our results show that the constructed codes are inherently self-orthogonal and, for m ≥ 7, also minimal. Self-orthogonal codes form an important class of linear codes with applications in design theory, lattice constructions, linear complementary dual (LCD) codes, communications, and cryptography. The families developed in this work provide new explicit examples of ternary self-orthogonal minimal codes, together with a complete characterization of their weight distributions. More generally, our approach offers a structural, quadratic-form-based alternative to existing function-driven methods for constructing self-orthogonal codes. It yields new explicit families of ternary self-orthogonal minimal codes, enriches the catalogue of p-ary few-weight codes, and introduces analytic tools that may extend to more general algebraic settings.

The Gram matrix is a classical object formed from the pairwise inner products of a collection of vectors, with fundamental roles in functional analysis, statistics, combinatorics, and coding theory. In the realm of sequence design, maximum-length sequences (m-sequences) are among the most fundamental classes of sequences, traditionally characterized by their span, decimation, shift-and-add, balance, run, and ideal autocorrelation properties. In this paper, we bridge the two foundational concepts by uncovering novel structural features of m-sequences through the lens of a family of Gram matrices. Specifically, for each 1 ≤ t ≤ 2^n - 1, we extract n consecutive subsequences of length t from an m-sequence of period 2^n - 1, construct their corresponding n × n Gram matrix, and investigate its rank, denoted by r_n(t). Utilizing semilinear representation of Galois groups and Bézoutian of polynomials, we derive an explicit formula for r_n(t) for all t, thereby establishing the complete rank distribution of these Gram matrices. Notably, we prove that full rank is attained for approximately half of the admissible values of t. We further uncover the intricate dynamics of r_n(t): rank-deficient states are strictly unstable (i.e., r_n(t) < n implies r_n(t+1) r_n(t)), whereas the full-rank state exhibits strong persistence, remaining at n over a nontrivial interval of consecutive values of t. Altogether, our results fully characterize both the global rank distribution and the local dynamics of rank function, as invariant of m-sequences. As an application, our findings completely determine the hull distribution of the family of punctured cyclic simplex codes.

Fractional repetition codes (FRCs)/Generalized FRCs (GFRCs) are classes of minimum bandwidth regenerating codes that play a key role in distributed storage systems. In this paper, first, we explore the inherent connection between biregular bipartite graphs and FRCs/GFRCs. Using the structural characteristics of these graphs, we determine the lower bound of the file size of the FRCs/GFRCs related to these graphs. Specifically, we leverage the eigenvalues of biregular bipartite graphs and derive several lower bounds on the file size of FRCs/GFRCs with flexible parameters and reconstruction degrees.

In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist both the Sidelnikov-Shestakov attack and the Wieschebrink attack. Constructing linear complementary dual (LCD) codes and determining the hull of linear codes have long been important topics in coding theory, as they play the crucial role in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), certain communication systems and cryptography. In this paper, by utilizing a class of non-GRS type linear codes, namely, generalized Roth-Lempel (in short, GRL) codes, we firstly construct several classes of Euclidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, generalized the main results given by Wu et al. in 2021. We also present an upper bound for the number of a class of Euclidean GRL codes with 1-dimensional hull, and then for several classes of Hermitian GRL codes, we firstly derive an upper bound for the dimension of the hull, and prove that the bound is attainable. Secondly, as an application, we obtain several families of EAQECCs. Thirdly, we prove that the GRL code is non-GRS for k >ℓ. Finally, some corresponding examples for LCD MDS codes and LCD NMDS codes are presented.

The binary asymmetric channel is a model for practical communication systems where the error probabilities for symbol transitions 0→ 1 and 1→0 differ substantially. In this paper, we introduce the notion of asymmetric Hamming bidistance (AHB) and its two-dimensional distribution, which separately captures directional discrepancies between codewords. This finer characterization enables a more discriminative analysis of decoding the error probabilities for maximum-likelihood decoding (MLD), particularly when conventional measures, such as weight distributions and existing discrepancy-based bounds, fail to distinguish code performance. Building on this concept, we derive a new upper bound on the average error probability for binary codes under MLD and show that, in general, it is incomparable with the two existing bounds derived by Cotardo and Ravagnani (IEEE Trans. Inf. Theory, 68 (5), 2022). To demonstrate its applicability, we compute the complete AHB distributions for several families of codes, including two-weight and three-weight projective codes (with the zero codeword removed) via strongly regular graphs and 3-class association schemes, as well as nonlinear codes constructed from symmetric balanced incomplete block designs (SBIBDs).

A uniform linear array (ULA) represents a fundamental sparse array configuration characterized by constant inter-element spacing. In this letter, for any positive integer r, we propose an rth-order generalized five-ULA sparse array (r-G5USA), which includes super augmented nested array (SANA) and a class of enhanced multi-ULA sparse array (EMUSA) structure as special cases. Closed-form expressions are derived for both the array configuration and the weight function of r-G5USA. Specifically, the proposed r-G5USA not only maintains a high number of uniform degrees of freedom (uDOFs) on the order of O(2N(2)/3 + 2N/3), but also enables flexible control of the first 2r-1 or 2r-2 weight functions, reducing their values to unity. Simulation results demonstrate that r-G5USA exhibits superior performance over existing sparse arrays in terms of uDOFs, coupling leakage (CL), and direction-of-arrival (DOA) estimation accuracy.

Subset sums over finite abelian groups lie at the intersection of additive combinatorics, design theory, and coding theory. Let G be a finite abelian group, and let _k^x be the family of k-subsets of G whose elements sum to x∈ G. This paper studies when the incidence structure (G,_k^x) is a block design. The elementary abelian p-group case was settled by Falcone and Pavone. Pavone (Des. Codes Cryptogr. 91 (2023), 2585–2603) further asked whether, for an arbitrary finite abelian group G, the zero-sum incidence structure (G,_k^0) can be a nontrivial 2-design only when G is an elementary abelian p-group. We settle this open question in the stronger form that, for every x∈ G, (G,_k^x) can be a nontrivial 2-design only if G is an elementary abelian p-group. The proof develops a character-theoretic approach to subset-sum designs, using character sums over the blocks to constrain the structure of the character group G. The approach also yields a complete characterization of subset-sum 1-designs and general arithmetic restrictions on subset-sum designs over arbitrary finite abelian groups, extending the corresponding results previously known for finite abelian p-groups.

. In this paper, we introduce a novel class of linear codes characterized by generator matrices whose column vectors are derived from two distinct regular hyperovals. We begin by providing a detailed characterization of NMDS (near Maximum Distance Separable) codes within this class. Building on this foundation, we determine the maximum length of these NMDS codes and enumerate the number of such codes achieving it, excluding equivalence. Next, we derive necessary and sufficient conditions for the equivalence of two linear codes in this class. Furthermore, we show that the automorphism group of these codes is isomorphic to the stabilizer subgroup of a subset of the finite field under the action of a specific subgroup of the semilinear group. Our construction guarantees at least a super-exponential number of inequivalent NMDS codes of maximum length. Lastly, for certain finite fields, we construct NMDS codes of maximum length from hyperovals using the Frobenius conjugacy classes.

In 2021, Jin et al. constructed four families of binary sequence sets via modified quadratic characters over finite fields. By analyzing rational points on elliptic curves and invoking the Hasse-Weil bound, they established that these sequence sets exhibit low correlation properties. In this paper, we propose an alternative approach: by transforming exponential sums involving quartic polynomials into those involving cubic polynomials and leveraging the more readily applicable Weil bound, we derive the same conclusions. Compared with their method, ours offers three key advantages: (i) it obviates the need for intricate analysis of elliptic curves, thereby streamlining technical procedures; (ii) it enables a more rigorous analysis of correlations; and (iii) it improves accessibility for researchers with limited expertise in algebraic geometry.

The design of error-correcting codes capable of correcting insertions and deletions has garnered significant attention recently, largely motivated by the requirements of DNA-based data storage. Insertion is one of the most frequent errors occurring in DNA sequences, where the inserted symbol is often identical or complementary to the original, and in practical implementations, noise can further cause the inserted symbol to mutate into a random one, which causes challenges to reliable data recovery. Motivated by these error mechanisms, this paper formalizes a noisy insertion channel characterized by an arbitrary number of identical or complementary symbol insertions, alongside at most one random insertion. Specifically, the exact coding capacity of this noisy insertion channel is established. By constructing asymptotically optimal error-correcting codes, this theoretical capacity is proven to be achievable. Furthermore, an efficient decoding algorithm is presented, which uniquely recovers the transmitted codewords in linear time with respect to the length of the received sequence.

Driven by the applications in DNA storage, codes capable of correcting errors in DNA sequences have attracted considerable interest. This paper concentrates on a scenario in which sequences are affected by errors known as reverse-complement duplications. More precisely, we construct errorcorrecting codes for channel where a single reverse-complement duplication of arbitrary length may happen. In general, handling reverse-complement duplications of variable (non-fixed) lengths involves combinatorial challenges, and our construction is the first code capable of correcting such errors of arbitrary lengths. By investigating some special combinatorial structures of the errors, we propose an explicit code construction using duplication-free sequences. Furthermore, we show that as the alphabet size q varies as an increasing function of n, the asymptotic rate is 1.

Frequency hopping sequences (FHSs) play a crucial role in frequency hopping (FH) communication systems due to their strong anti-interference ability, low interception probability, high confidentiality and strong concealment. The objective of this paper is to construct FHSs for quasi-synchronous frequency-hopping multiple access (FHMA) communication systems that simultaneously achieve optimal no-hit zone (NHZ) length and optimal gap. To accomplish this, the paper first derives tighter upper bounds for the gap size in both periodic and aperiodic scenarios under the assumption that all frequencies within the designated frequency slot set are fully utilized. Subsequently, this paper proposes a class of wide-gap frequency hopping sequences (WGFHSs) and a class of multi-timeslot wide-gap frequency hopping sequences (MTWGFHSs), both of which simultaneously exhibit optimal NHZ length and optimal gap.

Constructions of Z -complementary pairs (ZCPs) using generalized Boolean functions (GBFs) have a very rich literature. However, these works only focus on ZCPs having lengths with specific forms. In this paper, we propose a unified framework to construct q -phase ZCPs based on GBFs, which can obtain the algebraic structure of Type -I ZCPs of any even length. Through the proposed unified framework, we can propose several direct constructions of even -length ZCPs, which are not reported before. Also, all the previous constructions of Type -I ZCPs of even length based on GBFs, can be seen as special cases of the proposed construction.

In this paper, we establish the conditions for some finite abelian groups and the family all the $k$-sets in each of them summing up to an element $x$ to form $t$-designs. We fully characterize the sufficient and necessary conditions for the incidence structures to form $1$-designs in finite abelian $p$-groups, generalizing existing results on vector spaces over finite fields. For finite abelian groups of exponent $pq$, we also propose sufficient and necessary conditions for the incidence structures to form a $1$-designs. Furthermore, some interesting observations of the general case when the group is cyclic or non-cyclic are presented and the relations between $(t-1)$-designs and $t$-designs from subset sums are established. As an application, we demonstrate the correspondence between $t$-designs from the minimum-weight codewords in elliptic curve codes and subset-sum designs in their groups of rational points. By such a correspondence, elliptic curve codes supporting designs can be simply derived from subset sums in finite abelian groups that supporting designs.

The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding derivative decoding methods. One objective of this paper is to generalize these conclusions to q-ary extended cyclic codes with group algebra theory. It demonstrates that the cyclic DDs of a q-ary extended cyclic code are the same codes and its linear DDs are equivalent codes. In addition, we show that the relevant results can be generalized to q-ary constacyclic codes and the linear codes generated by Plotkin construction. Our conclusions reveal that the soft-decision decoding method proposed by Huang and Zhang for binary cyclic codes is also applicable to q-ary cyclic codes, q-ary constacyclic codes and the linear codes generated by Plotkin construction.

In this paper, we propose a direct construction of Golay complementary sets (GCSs) with flexible length aq m for set size q k using extended Boolean functions (EBFs), where a and q are positive integers with a <= q. As an extension of GCSs, we also propose two-dimensional (2-D) Golay complementary array sets (GCASs) using 2-D EBFs. The parameters of the proposed GCASs are more flexible as compared to the previous works.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes C f,g over F qm parameterized by any two functions f , g over F qm , and then study the properties of six types of C f,g , its punctured code C * f,g and their corresponding subfield codes over F q . The classification of C f,g is based on special choices of f , g as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of C f,g , we explicitly determine the weight distributions and dualities of C f,g , C* f,g and their subfield codes over F q . The remaining four types of C f,g are restricted to q = 2, and the weight distributions and dualities of the subfields code C (q) f,g and C * f,g (q) are completely determined. Most of the resultant linear codes (over F qm or over F q ) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [2 4m-2 , 2 m + 1, 2 4m-3 ] quaternary Hermitian self-orthogonal codes are obtained with m ≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

. In time-hopping multiple access ultra wide band (THMA-UWB) systems, time-hopping sequences (THSs) are crucial in determining the system's anti-jamming performance. We discuss the property of periodic cross correlation of THSs, which can be used to analytically evaluate the properties of THSs in codeword synchronism and codeword asynchronism but chip synchronism. Then we also construct several classes of nearly optimal or optimal THS sets with respect to the lower bound on the maximal cross correlation, which have longer sequence periods and more flexible family sizes than known ones. These THS sets may be more applicable for UWB systems. In particular, some proposed THSs are nearly optimal with respect to the Johnson bound on the maximal possible number of THSs.

Combinatorial neural (CN) codes are binary codes introduced firstly by Curto et al. for asymmetric channel, and then are further studied by Cotardo and Ravagnani under the metric δ _r (called asymmetric discrepancy) which measures the differentiation of codewords in CN codes. When r>1 , CN codes are different from the usual error-correcting codes in symmetric channel ( r=1 ). In this paper, we focus on the optimality of some CN codes with r>1 . An upper bound for the size of CN codes with δ _r=r+1 is deduced, by discussing the relationship between such CN codes and error-detecting codes for asymmetric channels, which is shown to be tight in this case. We also propose an improved Plotkin bound for CN codes. Notably, by applying symmetric designs related with Hadamard matrices, we not only generalize one former construction of optimal CN codes by bent functions obtained by Zhang et al. (IEEE Trans Inf Theory 69:5440–5448, 2023), but also obtain seven classes of new optimal CN codes meeting the improved Plotkin bound.