arXiv subjects

Huimin Lao

Publications and source records attributed to Huimin Lao.

10 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 $Γ(L,g,η)$ with $\operatorname{deg} g=t$ satisfies $d(Γ(L,g,η))=t+1$, and derive structured classes and infinite families attaining this distance.

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arXiv2026

Block Tensor Rank of Sum-Rank Metric Codes

Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its \emph{block tensor rank}: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the \emph{projection-wise bound} and the \emph{coordinate-code bound}. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the \emph{Singleton coordinate-code bound} and the \emph{Griesmer coordinate-code bound}, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

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arXiv2026

On the Minimum Distances of Some Families of Goppa Codes and BCH Codes

Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a necessary and sufficient criterion for a Goppa code to attain its designed distance $δ=t+1$, where $t$ is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of $q$-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with $G(x)=x^t+A$ from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case $G(x)=x^t$, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with $d=δ$, including the binary codes $\mathbf{C}_{(2,2^m-1,9,1)}$ and $\mathbf{C}_{(2,2^m-1,15,1)}$, the family $\mathbf{C}_{(p,p^p-1,2p+2,1)}$ with an odd prime $p$ and the family $\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$ with $r\mid q-1$. In particular, we prove that the primitive BCH code $\mathbf{C}_{(q,q^m-1,q^t+1,1)}$ has minimum distance $q^t+1$ under the condition $t\mid m$, improving the previously known condition $pt\mid m$.

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arXiv2026

On the Minimum Distances of Some Families of BCH Codes

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes with $d=δ$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $δ$, and offset $b$, denoted by $\mathbf{C}_{(q, n, δ, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=δ$, where $δ\in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=δ$, where $2 \le δ\le q-1$. For $δ=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=δ$ for all $m$ satisfying $m \equiv 0 \pmod{pt}$, where $p$ denotes the characteristic of $\mathbb{F}_q$. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ is always equal to its Bose distance $d_B$. Our result confirms this conjecture for the case $m \equiv 0 \pmod{pt}$. For non-primitive BCH codes, we construct a family of BCH codes $\mathbf{C}_{(q,\frac{q^p-1}λ,p+1,1)}$ with $d=δ=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $λ\mid q-1$.

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arXiv2026

Concatenated Sum-Rank Codes

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vladut-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vladut-Zink-like bound and the Gilbert-Varshamov-like bound.

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arXiv2025

On de Bruijn Array Codes Part II: Linear Codes

An M-sequence generated by a primitive polynomial has many interesting and desirable properties. A pseudo-random array is the two-dimensional generalization of an M-sequence. There are non-primitive polynomials all of whose non-zero sequences have the same period. These polynomials generate \emph{sets} of sequences with properties similar to M-sequences. In this paper, a two-dimensional generalization for such sequences is given. This generalization is for a pseudo-random array code, which is a set of $r_1 \times r_2$ arrays in which each $n_1 \times n_2$ nonzero matrix is contained exactly once as a window in one of the arrays. Moreover, these arrays have the shift-and-add property, i.e., the bitwise addition of two arrays (or a nontrivial shift of such arrays) is another array (or a shift of another array) from the code. All the known arrays can be formed by folding sequences generated from an irreducible polynomial or a reducible polynomial whose factors have the same degree and the same exponent. Two proof techniques are used to prove the constructions are indeed of pseudo-random array codes. The first technique is based on another method, different from folding, for constructing some of these arrays. The second technique is a generalization of a known proof technique. This generalization enables the construction of pseudo-random arrays with parameters not known before, and also provides a variety of pseudo-random array codes which cannot be generated by the first method. The two techniques also suggest two different hierarchies between pseudo-random array codes. Finally, two methods to verify whether a folding of sequences, generated by these polynomials, yields a pseudo-random array or a pseudo-random array code, will be presented.

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arXiv2025

On optimal quantum LRCs from the Hermitian construction and $t$-designs

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 $t\in \{2,3\}$, 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.

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arXiv2025

Properties and Decoding of Twisted GRS Codes and Their Extensions

Maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes. Alongside near MDS (NMDS) codes, they are applicable in communication, cryptography, and storage systems. From theoretical perspective, it is particularly intriguing to investigate families of linear codes in which each element can be determined to be either a non-GRS MDS or an NMDS code. Two promising candidates for such families emerge from what is known as twisted GRS (TGRS) construction. These candidates are the $(+)$-TGRS codes and their extended versions, called $(+)$-extended TGRS (ETGRS) codes. Although many of their properties have been characterized, there are gaps to fill. Which among the codes are non-GRS MDS? Can we improve on their decoding by using their error-correcting pairs or deep holes? In this paper we solve these problems. The answer to the first problem leads us to two classes of non-GRS MDS Hermitian self-dual TGRS codes and a proof that there is no Galois self-dual ETGRS code. Addressing the second problem, we present an explicit decoding algorithm for ETGRS codes that outperforms existing decoding algorithms given some conditions. By considering the duals of TGRS codes which are MDS, we determine the covering radius and a class of deep holes of the recently constructed non-GRS MDS codes due to Han and Zhang.

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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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arXiv2020

Parameter-controlled inserting constructions of constant dimension subspace codes

A basic problem in constant dimension subspace coding is to determine the maximal possible size ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in ${\bf F}_q^n$ such that the subspace distance satisfies $\operatorname{dis}(U,V)=2k-2\dim(U \cap V) \geq d$ for any two different $k$-dimensional subspaces $U$ and $V$ in this set. In this paper we propose new parameter-controlled inserting constructions of constant dimension subspace codes. These inserting constructions are flexible because they are controlled by parameters. Several new better lower bounds which are better than all previously constructive lower bounds can be derived from our flexible inserting constructions. $141$ new constant dimension subspace codes of distances $4,6,8$ better than previously best known codes are constructed.

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