Hengfeng Liu

Hengfeng Liu, Chunming Tang, Zhengchun Zhou et al.

A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t > 2$, all with length at most $q + 1$. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than $q + 1$ are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first \emph{generic construction} of $q$-ary NMDS codes supporting 2-designs with lengths \emph{exceeding $q + 1$}. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

Hengfeng Liu, Sihem Mesnager, Chunming Tang et al.

Locally recoverable codes (LRCs) have emerged as fundamental objects in modern coding theory, primarily due to their pivotal role in distributed and cloud storage systems. A major breakthrough in their construction was achieved by Tamo and Barg, who introduced the notion of \emph{good polynomials} as a key structural ingredient. In this article, we propose a natural generalization of this paradigm by introducing the concept of \emph{good rational functions}. Building upon this extension, we develop a unified and flexible framework for constructing optimal LRCs. To quantify the quality of a rational function, we embed the problem into the rich context of algebraic function field theory and Galois theory. This perspective allows us to extend the Galois-theoretic framework originally developed by Micheli for good polynomials. In particular, we derive structural and quantitative results on the number of totally split rational places associated with rational functions. Furthermore, we construct explicit families of good rational functions that outperform all good polynomials of the same degree. As a consequence, we obtain infinite families of optimal LRCs with improved parameters compared to those arising from the classical Tamo-Barg construction. These results highlight the intrinsic strength of our approach.

Hengfeng Liu, Chunming Tang, Cuiling Fan et al.

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 $1 \le t \le 2^n - 1$, we extract $n$ consecutive subsequences of length $t$ from an m-sequence of period $2^n - 1$, construct their corresponding $n \times n$ 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 $r_n(t+1) \ne r_n(t)$), 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.

Hengfeng Liu, Chunming Tang, Cuiling Fan

DNA storage has emerged as a promising solution for large-scale and long-term data preservation. Among various error types, insertions are 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 creates significant challenges to reliable data recovery. In this paper, we investigate a new noisy insertion channel, where infinitely many insertions of symbols complement or identical to the original ones and up to one insertion of random symbol may occur. We determine the coding capacity of the noisy channel and construct asymptotically optimal error-correcting codes achieving the coding capacity.