We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of ^⊥_ H. Every [n+1,k+1]_q^2 linear code with d(^⊥_ H)>1 is monomially equivalent to the generalized extended code ( u,a) of an [n,k]_q^2 linear code for a fixed a∈_q^2^* and some u∈_q^2^n. We then characterize the Hermitian hull and Hermitian dual distance of ( u,a) in terms of the position of u relative to +^⊥_ H and the interaction between u and the minimum weight codewords of ^⊥_ H, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of ( u,a). In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of ( u,a) are derived. Applying these results to the Hermitian construction for EAQECCs gives us 267 new EA qubit codes of lengths n ≤ 40 and 14 new EA qutrit codes of lengths n ≤ 25 compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for 236 qubit and 8 qutrit codes.

The error coefficient of a linear code, defined as the number of minimum-weight codewords, plays a central role in evaluating the performance of the code. In this paper, we establish two recursive bounds on the minimum possible error coefficient among optimal linear codes with prescribed parameters. We prove that these bounds are tight in infinitely many cases by constructing two explicit infinite families of optimal linear codes that attain them with equality, and we further show that MDS codes also meet one of the proposed bounds with equality. Beyond the recursive-bound framework, we determine the minimum possible error coefficient for three explicit families of optimal linear codes: two families arising from simplex codes and one family associated with MacDonald codes. Moreover, employing tools from combinatorial design theory, we solve a problem proposed by Guan et al. [12] on the eventual constancy of the minimum error coefficient of optimal codes.

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. A linear code C is called h_ℓ -linear if C has ℓ -dimensional hull. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to h_ℓ -linear codes with ℓ≥ 1 . Therefore, the objective of this paper is to investigate an interesting but non-trivial problem, which is to study some properties of binary h_1 -linear codes and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. Furthermore, we study the largest minimum distance d_one(n,k) among all binary h_1 -linear [n, k] codes. We determine the largest minimum distances d_one(n,n-k) for k≤ 5 and d_one(n,k) for k≤ 4 or 14≤ n≤ 24 . We partially determine the exact value of d_one(n,k) for k=5 or 25≤ n≤ 30 .

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo et al. (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for t∈{2,3}, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

The existence of optimal binary self-orthogonal codes has been well characterized. In this paper, we develop general methods involving residual codes and the MacWilliams identities to prove the nonexistence of several infinite families of binary self-orthogonal codes, despite the existence of binary linear codes with the same parameters. In particular, we focus on the largest minimum distances of optimal binary self-orthogonal codes with dimension eight.

The first objective of this paper is to characterize all possible parameters of Plotkin-optimal two-homogeneous weight regular projective codes over finite chain rings, as well as their weight distributions. We show the existence of codes with these parameters by constructing an infinite family of two-homogeneous weight codes. The parameters of their Gray images have the same weight distribution as that of the two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull. Lond. Math. Soc., 18 (1986) 97-122). Further, we also construct three-homogeneous weight regular projective codes over finite chain rings combining with some known results. Finally, we study applications of our constructed codes in secret sharing schemes and graph theory. In particular, infinite families of strongly regular graphs and strongly walk-regular graphs with non-trivial parameters are obtained.

The quantum chromatic number is a fundamental parameter in the study of nonlocal games, capturing the extent to which entanglement can improve performance in distributed tasks. In this paper, we investigate the quantum chromatic number of generalized Johnson graphs. By constructing modulus-one orthogonal representations, we obtain general upper bounds on their quantum chromatic numbers. We further analyze the smallest eigenvalue of these graphs. Combining the resulting Hoffman-type lower bounds with the upper bounds obtained from orthogonal representations, we determine the exact quantum chromatic numbers of two infinite families of generalized Johnson graphs. Finally, applying a forbidden-distance theorem for binary codes, we show that the classical chromatic numbers of these families grow exponentially with n, whereas their quantum chromatic numbers grow linearly. These families exhibit an exponential separation between the classical and quantum chromatic numbers.

The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the b-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the b-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the b-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the b-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters [n, k, dH(C)]q of cyclic codes is provided. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The hull of a linear code is the intersection of the code and its dual code, which is effective for determining parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). There are few constructions of linear codes with various Hermitian hull dimensions, aside from Hermitian LCD and self-orthogonal codes. The object of this paper is to introduce a buildingup construction for constructing linear [n + 2, k + 1] codes with l or (l + 1)dimensional Hermitian hull from a given linear [n, k] code with l-dimension Hermitian hull of a smaller length. This construction includes the converse of the famous shortening technique as a special case. Using this method, we construct optimal quaternary linear codes of lengths up to 13 with Hermitian hull dimensions 2-5. As an application, we construct many EAQECCs, which improve the parameters of EAQECCs of Grassl's code table.

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed 5. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to 2.

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing Solomon-Stiffler codes and some related residual codes. Second, using such a characterization, we determine the exact value of $d_{so}(n,7)$ except for five special cases and the exact value of $d_{so}(n,8)$ except for 41 special cases, where $d_{so}(n,k)$ denotes the largest minimum distance among all binary self-orthogonal $[n, k]$ codes. Currently, the exact value of $d_{so}(n,k)$ $(k \le 6)$ was determined by Shi et al. (2022). In addition, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code.

For an integer s >= 1, let C-s(q(0)) be the generalized Zetterberg code of length q(0)(s) + 1 over the finite field F-q0 of odd characteristic. Recently, Shi et al. determined the covering radius of C-s(q(0)) for q(0)(s)not equivalent to 7(mod 8), and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for q(0)(s)equivalent to 7(mod 8), which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length q(0)(s)+1/2, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

The existence of $q$-ary linear complementary pairs (LCPs) of codes with $q> 2$ has been completely characterized so far. This paper gives a characterization for the existence of binary LCPs of codes. As a result, we solve an open problem proposed by Carlet $et~al.$ (IEEE Trans. Inf. Theory 65(3): 1694-1704, 2019) and a conjecture proposed by Choi $et~al.$ (Cryptogr. Commun. 15(2): 469-486, 2023).

A binary linear code is called asymptotic frame error rate (AFER)-optimal if it achieves the maximum possible value of the minimum distance while having the smallest value of the corresponding error coefficient. Over the additive white Gaussian noise channel and under maximum-likelihood decoding, AFER-optimal codes attain the best possible asymptotic frame error rate at high signal-to-noise ratio. In this paper, we present several bounds on the smallest error coefficients of binary linear codes and give several constructions of AFER-optimal binary linear codes. Many examples confirm that our bounds are sharp on numerous occasions. In addition, we give two families of AFER-optimal codes that respectively attain the proposed bounds with equality.

The hull of a linear code over a finite field is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main objective of this paper is to obtain a closed mass formula for linear codes with prescribed hull dimension. We simplify the mass formula obtained by Sendrier and provide an alternative proof for the mass formula for self-orthogonal codes obtained by Pless. Finally, we obtain a classification of (optimal) ternary linear codes with small parameters.

By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.

Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kløve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over ${\mathbb {F}}_{q^{\ell } }$ of a q-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.

Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information protection. In this paper, we give some methods for constructing LCD codes over small finite fields by modifying some typical methods for constructing linear codes. We show that all odd-like binary Euclidean LCD codes, ternary Euclidean LCD codes and quaternary Hermitian LCD codes can be constructed using the modified methods. Our results improve the known lower bounds on the largest minimum distances of LCD codes. Furthermore, we give two counterexamples to disprove the conjecture proposed by Bouyuklieva (Des. Codes Cryptogr. 89(11), 2445–2461 2021).

The hull of a linear code over finite fileds is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main purpose of this paper is to obtain the closed mass formula for binary linear codes with various hull dimensions, which simplifies the mass formula obtained by Sendrier in (SIAM J. Discrete Math., 10(2): 282-293, 1997). We show that almost all binary linear codes with ℓ-dimensional hull are odd-like codes with odd-like duals for fixed ℓ. We also study the largest minimum distance of a binary linear [n, k] code with ℓ-dimensional hull. Most importantly, we give a complete classification of binary linear codes with various hull dimensions for n ≤ 12 using a building-up construction, which is confirmed by double-checking with our mass formula. We also give the classification of optimal binary linear [n, k] codes with various hull dimensions for n ≤ 13. Combining with known results, we obtain the classification of (optimal) binary linear codes with small parameters.

In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via ℓ-intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform the best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good.