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.

For integersu≥ 2 ands≥ 1, letq0 = 2u, and letCs(q0) be the generalized Zetterberg code of lengthn= qs0 + 1 over the finite field Fq0of characteristic 2. For odd characteristic, the covering radius ofCs(q0) was determined recently, whereas the case of even characteristic remained open. In this paper, we determine the covering radius of generalized Zetterberg codes over finite fields of characteristic 2, thereby solving this open problem. Our approach uses methods from the theory of algebraic curves over finite fields. As an application, we obtain an infinite family of quasi-perfect codes.

Group codes form an important class of spherical codes, consisting of a single orbit under a subgroup of the orthogonal group. They have been studied by (Mittelholzer-Lahtonen, 1996) in the case of Coxeter groups for their packing radius. We study them here for the same groups, with respect to their covering radius. An exact algorithm to determine their covering radii is derived, based on geometric ideas, and applied to tabulate their values in dimensions up to 8. The values of the covering radii are compared to the sphere covering bound, and to the bounds in (Fazekas-Levenshtein, 1995) and (Boyvalenkov-Stoyanova, 2021) on the covering radius of spherical designs with given strength. For many sizes of codes, our codes are reasonably sparse sphere coverings, and the only known in these dimensions.

A binary linear code is calledasymptotic frame error rate (AFER)-optimalif 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.

Carlet et al. showed that for$q\gt 2$, there exists a q-ary linear complementary pair (LCP) of codes whose security parameter is as good as the minimum distance of the best linear code with the same length and dimension. In this paper, we study the best security parameters 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).

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.

Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing schemes, authentication codes and association schemes. There are two major ingredients in this paper. The first ingredient is to solve four conjectures recently proposed by Heng and Ding (IEEE Trans. Inf. Theory, 68(6): 3643-3656, 2022). Besides, we also determine the weight distributions of subfield codes derived from$x^{3}$and oval polynomials over${\mathbb {F}}_{2^{m}}$. The second ingredient is to obtain two classes ofp-ary linear codes from monomials over${\mathbb {F}}_{p^{m}}$. We study the parameters of the constructed codes and determine their weight distributions. Notably, we show that the codes$\mathcal {C}_{(x^{p^{i}},p^{m})}^{(p)}$are optimal and almost optimal in many cases with respect to the online Database of Grassl. Finally, we observe that the derived linear codes also have the minimality property for most cases.

For an integer$s\ge 1$, let${\mathcal {C}}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over the finite field${\mathbb {F}}_{q_{0}}$of odd characteristic. Recently, Shi et al. determined the covering radius of${\mathcal {C}}_{s}(q_{0})$for$q_{0}^{s} \cancel {\equiv }7 \pmod {8}$, 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$q_{0}^{s} \equiv 7 \pmod {8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length$\frac {q_{0}^{s}+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

We introduce the notion of logarithmically concave (or log-concave) sequences in coding theory. A sequencea0,a1, . . . ,anof real numbers is called log-concave ifa2i⩾ai−1ai+1for all 1 ⩽i⩽n− 1. A natural sequence of positive numbers in coding theory is the weight distribution of a linear code consisting of the nonzero values among Ai’s where Ai denotes the number of codewords of weighti. We call a linear code log-concave if its nonzero weight distribution is log-concave. Our main contribution is to show that all binary general Hamming codes of length 2r−1 (r= 3 orr⩾ 5), the binary extended Hamming codes of length 2r(r⩾ 3), and the second order Reed-Muller codesR(2,m) (m⩾ 2) are all log-concave while the homogeneous and projective second order Reed-Muller codes are either log-concave, or 1-gap log-concave. Furthermore, we show that any MDS [n, k] code over Fqsatisfying 3 ⩽k⩽n/2 + 3 is log-concave ifq⩾q0(n, k) which is the larger root of a quadratic polynomial. We also show that most of QR codes, BCH codes and Roth-Lempel NMDS codes are not log-concave. Hence, we expect that the concept of log-concavity in coding theory will stimulate many interesting problems.

The symbol-pair coding theory was put forward by Cassuto and Blaum [IEEE TIT, 2011] to be applicable in high-density storage situations. Yaakobiet al. [IEEE TIT, 2016] extended the concept of symbol-pair metric tob-symbol metric whenb≥ 2. The extensive research on theb-symbol Hamming weight spectra of cyclic codes has been centered on the case where the alphabet is a finite field. The case of cyclic codes over Z4, an extremely important class of codes, has been overlooked for a long time in the exploration of theb-symbol Hamming weight spectra. In this paper, we study theb-symbol Hamming weight spectra of the shortened Kerdock codesK−mand the Kerdock codesKmover Z4. The formulas for calculating the symbol-pair Hamming weight of the codewords inK−mandKmare given, and their values hinge on the trace-like function values of specific elements in the Teichmüller set. In particular, we present a class of Z4-cyclic codes with three non-zerob-symbol Hamming weights. As by-products, theb-symbol Hamming weight hierarchies of the Preparata codes and the Goethals codes are provided.

Federated Learning (FL) presents an innovative approach to privacy-preserving distributed machine learning and enables efficient crowd intelligence on a large scale. However, a significant challenge arises when coordinating FL with crowd intelligence which diverse client groups possess disparate objectives due to data heterogeneity or distinct tasks. To address this challenge, we propose the Federated cINN Clustering Algorithm (FCCA) to robustly cluster clients into different groups, avoiding mutual interference between clients with data heterogeneity, and thereby enhancing the performance of the global model. Specifically, FCCA utilizes a global encoder to transform each client’s private data into multivariate Gaussian distributions. It then employs a generative model to learn encoded latent features through maximum likelihood estimation, which eases optimization and avoids mode collapse. Finally, the central server collects converged local models to approximate similarities between clients and thus partition them into distinct clusters. Extensive experimental results demonstrate FCCA’s superiority over other state-of-the-art clustered federated learning algorithms, evaluated on various models and datasets. These results suggest that our approach has substantial potential to enhance the efficiency and accuracy of real-world federated learning tasks.

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.

A code$C$is called$Z_{p}Z_{p^{2}}$-linear if it is the Gray image of a$Z_{p}Z_{p^{2}}$-additive code. For any prime number$p$larger than 3, the bounds of the rank of$Z_{p}Z_{p^{2}}$-linear codes are given. For each value of the rank and the pairs of rank and the dimension of the kernel of$Z_{p}Z_{p^{2}}$-linear codes, we give detailed construction of the corresponding codes. As an example, the rank and the dimension of the kernel of$Z_{5}Z_{25}$-linear codes are studied.

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.

Let$ {\mathbb F}_{q_{0}}$be a finite field of odd characteristic. For an integer$s\ge 1$, let$\mathcal {C}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over$ {\mathbb F}_{q_{0}}$. If$s$is even, then we prove that the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. Put$q=q_{0}^{s}$. If$s$is odd and$q \not \equiv 7 \mod 8$, then we present an explicit lower bound$N_{1}(q_{0})$so that if$s \ge N_{1}(q_{0})$, then the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. We also show that the covering radius of$\mathcal {C}_{1}(q_{0})$is 2. Moreover we study some cases when$s$is an odd integer with$3 \le s \le N_{1}(q_{0})$and, rather unexpectedly, we present concrete examples with covering radius 2 in that range. We introduce half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 1 \mod 4$. Similarly we introduce twisted half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 3 \mod 4$. We show that the same results hold for the half and twisted half generalized Zetterberg codes.

The purpose of this paper is to solve the two conjectures on the largest minimum distance$d_{so}(n,5)$of a binary self-orthogonal$[n, 5]$code proposed by Kim and Choi (2022). The determination of$d_{so}(n,k)$has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension$k$increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal$[n,k]$codes were constructed for$k=4,5$. Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on$d_{so}(n,5)$. In this paper, we develop a general method to determine the exact value of$d_{so}(n,k)$for$k=5,6$and show that the two conjectures made by Kim and Choi (2022) are true.

The Galois ring GR$(4^{\Delta})$is the residue ring$Z_{4}[x]/(h(x))$, where$h(x)$is a basic primitive polynomial of degree$\Delta $over$Z_{4}$. For any odd$\Delta $larger than 1, we construct a partition of GR$(4^{\Delta}) \backslash \{0\}$into 6-subsets of type$\{a,b,-a-b,-a,-b,a+b\}$and 3-subsets of type$\{c,-c,2c\}$such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^{\Delta})$and, if$\Delta $is not a multiple of 3, under the action of the automorphism group of GR$(4^{\Delta})$. As a corollary, this implies the existence of quasi-cyclic additive 1-perfect codes of index$(2^{\Delta} -1)$in$D((2^{\Delta} -1)(2^{\Delta} -2)/{6}, 2^{\Delta} -1)$where$D(m,n)$is the Doob metric scheme on$Z^{2m+n}$.

The$r$-th generalized Hamming metric and the$b$-symbol metric are two different generalizations of Hamming metric. The former is used on the wire-tap channel of Type II, and the latter is motivated by the limitations of the reading process in high-density data storage systems and applied to a read channel that outputs overlapping symbols. In this paper, we study the connections among the three metrics (that is, Hamming metric,$b$-symbol metric, and$r$-th generalized Hamming metric) mentioned above and give a conjecture about the$b$-symbol Griesmer Bound for cyclic codes.

We prove that the covering radius of the Melas code$M(m,q)$of length$n=q^{m}-1$over$\mathbb {F}_{q}$is 2 if$q > 3$. We also prove that the covering radius of$M(m,3)$is 3 is$m \ge 3$, the covering radius of$M(2,3)$is 4, and the covering radii of$M(1,2)$and$M(1,3)$are 1.

We study codes with parameters of$q$-ary shortened Hamming codes, i.e.,$(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal$q$-fold packing with minimum distance 2. Secondly, for every admissible length starting from$n=20$, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper.

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.

We investigate fat projective linear codes over${\mathbb Z}_{p^{m}}$,$m\geqslant 2$, with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry$\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry.

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the$\mathbb {Z}_{2} \mathbb {Z}_{4}$-additive quasi-cyclic codes.

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and \mathbb Fqis an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n-1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q.

The dual of the Kasami code of length q2- 1, with q a power of 2, is constructed by concatenating a cyclic MDS code of length q + 1 over Fq with a Simplex code of length q - 1. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related q-ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code.

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function Γ(k, q), that is defined as the largest number of nonzero weights a cyclic code of dimension k over Fq can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or q-ary, as well as in the q-ary Hamming code are determined, two difficult results of independent interest.

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function ΓQ(n, ℓ, k, q), that is defined to be the largest number of nonzero weights a quasi-cyclic code of index gcd(ℓ, n), length n and dimension k over Fqcan have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a q-ary Reed-Muller code is quasi-cyclic, a result of independent interest.

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on $$2^{10}$$ vertices, with new intersection array $$\{33,30,15;1,2,15\}$$ . The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters $$(2^{10}, 495,238, 240)$$ . Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed.

The article “A new distance-regular graph of diameter 3 on 1024 vertices", written by Minjia Shi, Denis S. Krotov and Patrick Solé, was originally published electronically on the publisher's internet portal (currently SpringerLink) on 24 January 2019 without open access.

In this paper, two different Gray-like maps from Zpα× Zpkβ, where p is prime, to Zpn, n = α t βpk-1, denoted by φ and Φ, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if C is a ZpZpk-additive code, and C⊥is its dual, then the weight enumerators of the image p-ary codes φ(C) and Φ(C⊥) are formally dual. This is a partial generalization of [D. S. Krotov, On Z2k-dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed ZpZp2... Zpk alphabet is given.

We construct two new infinite families of trace codes of dimension 2m, over the ring Fp+ uFp, with u2= u, when p is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear p-ary codes of respective lengths (pm-1)2and 2(pm-1)2. When m is singly even, the first family gives five-weight codes. When m is odd and p ≡ 3 (mod 4), the first family yields p-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever p = 3 and m ≥ 3, or p ≥ 5 and m ≥ 4. Applications to secret sharing schemes are given.

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over Z4with type 4k12k2.

In this paper, we investigate the structure and properties of cyclic codes over the ringF2+vF2. We first study the relationship between cyclic codes overF2+vF2and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given lengthnoverF2+vF2.

In this paper, we investigate the structure and properties of cyclic codes over the ring F2+ vF2. We first study the relationship between cyclic codes over F2+ vF2and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length over F2+ vF2.