arXiv subjects

Conghui Xie

Publications and source records attributed to Conghui Xie.

8 recordsLinked to original sources

arXiv2026

Generalized BCH Codes and Twisted Goppa Codes Attaining Their Designed Distances

Determining the true minimum distance of an alternant code remains a notoriously difficult problem in coding theory. In this paper, we study the minimum distances of generalized BCH codes and twisted Goppa codes through their parity-check matrices. We first give a necessary and sufficient condition for an alternant code to attain its designed distance and apply it to generalized BCH codes. As applications, we prove that broad classes of generalized BCH codes have minimum distances equal to their designed distances. These classes provide explicit infinite families rather than isolated examples. We characterize when a twisted Goppa code $\Gamma(L,g,\eta)$ with $\operatorname{deg} g=t$ satisfies $d(\Gamma(L,g,\eta))=t+1$, and derive structured classes and infinite families attaining this distance.

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arXiv2025

$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes

Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is stronger than the classical result of Beutelspacher in 1983. Then a lower bound on lengths of projective $s$-minimal codes is also obtained. It is proved that $(s+1)$-minimal codes are certainly $s$-minimal codes. We generalize the Ashikhmin-Barg condition for minimal codes to $s$-minimal codes. Many infinite families of $s$-minimal codes satisfying and violating this generalized Ashikhmin-Barg condition are constructed. We also give several examples which are binary minimal codes, but not $2$-minimal codes.

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arXiv2025

Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing bound have been reported in the literature. However, it is a 75-year-old open problem in coding theory whether there is an infinite family of distance-optimal linear codes over finite fields with arbitrarily large minimum distance with respect to the sphere packing bound. This main objective of this paper is to settle this long-standing open problem in coding theory. As by-products, several infinite families of distance-optimal binary codes with small minimum distances are presented. Two infinite families of binary five-weight codes are reported. Some open problems are also proposed.

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arXiv2025

Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes

In recent years, there have been many constructions of minimal linear codes violating the Ashikhmin-Barg condition from Boolean functions, linear codes with few nonzero weights or partial difference sets. In this paper, we first give a general method to transform a minimal code satisfying the Ashikhmin-Barg condition to a minimal code violating the Ashikhmin-Barg condition. Then we give a construction of a minimal code satisfying the Ashikhmin-Barg condition from an arbitrary projective linear code. Hence an arbitrary projective linear code can be transformed to a minimal codes violating the Ashikhmin-Barg condition. Then we give infinite many families of minimal codes violating the Ashikhamin-Barg condition. Weight distributions of constructed minimal codes violating the Ashikhmin-Barg condition in this paper are determined. Many minimal linear codes violating the Ashikhmin-Barg condition with their minimum weights close to the optimal or the best known minimum weights of linear codes are constructed in this paper. Moreover, many infinite families of self-orthogonal binary minimal codes violating the Ashikhmin-Barg condition are also given.

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arXiv2024

Linear Codes from Projective Linear Anticodes Revisited

An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $\delta$ is a code in ${\bf F}_q^n$ such that the distance between any two distinct codewords in ${\bf C}$ is at most $\delta$. The famous Erd\"{o}s-Kleitman bound for a binary anticode ${\bf C}$ of the length $n$ and the diameter $\delta$ asserts that $$|{\bf C}| \leq \Sigma_{i=0}^{\frac{\delta}{2}} \displaystyle{n \choose i}.$$ In this paper, we give an antiGriesmer bound for $q$-ary projective linear anticodes, which is stronger than the above Erd\"{o}s-Kleitman bound for binary anticodes. The antiGriesmer bound is a lower bound on diameters of projective linear anticodes. From some known projective linear anticodes, we construct some linear codes with optimal or near optimal minimum distances. A complementary theorem constructing infinitely many new projective linear $(t+1)$-weight code from a known $t$-weight linear code is presented. Then many new optimal or almost optimal few-weight linear codes are given and their weight distributions are determined. As a by-product, we also construct several infinite families of three-weight binary linear codes, which lead to $l$-strongly regular graphs for each odd integer $l \geq 3$.

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arXiv2024

Two classes of constacyclic codes with a square-root-like lower bound

Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of $q$-ary negacyclic codes of length $(q^m-1)/2$ and an infinite class of $q$-ary constacyclic codes of length $(q^m-1)/(q-1)$ are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.

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arXiv2024

Repeated-Root Cyclic Codes with Optimal Parameters or Best Parameters Known

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary and $p$-ary cyclic codes for odd prime $p$ in the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over $\bF_q$ for each even $q \geq 4$. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters $[2n, k, d \geq (n-1)/\log_2 n]$, where $n=2^m-1$ and $k \geq n$, are constructed. In addition, 27 repeated-root cyclic codes of length up to $254$ over $\bF_q$ for $q \in \{2, 4, 8\}$ with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

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arXiv2024

Cyclic and Negacyclic Sum-Rank Codes

Sum-rank codes have known applications in the multishot network coding, the distributed storage and the construction of space-time codes. U. Mart\'{\i}nez-Pe\~{n}as introduced the cyclic-skew-cyclic sum-rank codes and proposed the BCH bound on the cyclic-skew-cyclic sum-rank codes in his paper published in IEEE Trans. Inf. Theory, vol. 67, no. 8, 2021. Afterwards, many sum-rank BCH codes with lower bounds on their dimensions and minimum sum-rank distances were constructed. Sum-rank Hartmann-Tzeng bound and sum-rank Roos bound on cyclic-skew-cyclic codes were proposed and proved by G. N. Alfarano, F. J. Lobillo, A. Neri, and A. Wachter-Zeh in 2022. In this paper, cyclic, negacyclic and constacyclic sum-rank codes are introduced and a direct construction of cyclic, negacyclic and constacyclic sum-rank codes of the matrix size $m \times m$ from cyclic, negacyclic and constacyclic codes over ${\bf F}_{q^m}$ in the Hamming metric is proposed. The cyclic-skew-cylic sum-rank codes are special cyclic sum-rank codes. In addition, BCH and Hartmann-Tzeng bounds for a type of cyclic sum-rank codes are developed. Specific constructions of cyclic, negacyclic and constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances are proposed. Moreover, many distance-optimal binary sum-rank codes and an infinite family of distance-optimal binary cyclic sum-rank codes with minimum sum-rank distance four are constructed. This is the first infinite family of distance-optimal sum-rank codes with minimum sum-rank distance four in the literature.

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