Chunming Tang

Can Xiang, Chunming Tang

Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields and present a construction method of cyclic MDS codes over finite fields. We transform the problem of verifying the MDS property over a finite field into a problem of determining non-zero minors in characteristic zero. Compared with existing construction methods, our method is relatively simple. In particular, the results of this paper show that the parameters of non-RS cyclic MDS codes are flexible and completely cover the results in [Non-Reed-Solomon Type Cyclic MDS codes, IEEE Trans. Inf. Theory, 71(5): 3489--3496, 2025].

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.

Shukai Wang, Cuiling Fan, Chunming Tang et al.

The binary asymmetric channel is a model for practical communication systems where the error probabilities for symbol transitions $0\rightarrow 1$ and $1\rightarrow0$ differ substantially. In this paper, we introduce the notion of asymmetric Hamming bidistance (AHB) and its two-dimensional distribution, which separately captures directional discrepancies between codewords. This finer characterization enables a more discriminative analysis of decoding the error probabilities for maximum-likelihood decoding (MLD), particularly when conventional measures, such as weight distributions and existing discrepancy-based bounds, fail to distinguish code performance. Building on this concept, we derive a new upper bound on the average error probability for binary codes under MLD and show that, in general, it is incomparable with the two existing bounds derived by Cotardo and Ravagnani (IEEE Trans. Inf. Theory, 68 (5), 2022). To demonstrate its applicability, we compute the complete AHB distributions for several families of codes, including two-weight and three-weight projective codes (with the zero codeword removed) via strongly regular graphs and 3-class association schemes, as well as nonlinear codes constructed from symmetric balanced incomplete block designs (SBIBDs).

Sihem Mesnager, Chunming Tang, Maosheng Xiong

At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack (which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers). Next, Boura and Canteaut introduced an important parameter (related to the BCT) for cryptographic Sboxes called boomerang uniformity. In this context, we present a brief state-of-the-art on the notion of boomerang uniformity of vectorial functions (or Sboxes) and provide new results. More specifically, we present a slightly different (and more convenient) formulation of the boomerang uniformity and show that the row sum and the column sum of the boomerang connectivity table can be expressed in terms of the zeros of the second-order derivative of the permutation or its inverse. Most importantly, we specialize our study of boomerang uniformity to quadratic permutations in even dimension and generalize the previous results on quadratic permutation with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity). As a consequence of our general result, we prove that the boomerang uniformity of the binomial differentially $4$-uniform permutations presented by Bracken, Tan, and Tan equals $4$. This result gives rise to a new family of optimal Sboxes.

Zongxiang Yi, Yuyin Yu, Chunming Tang et al.

Notes on two constructions of zero-difference balanced (ZDB) functions are made in this letter. Then ZDB functions over $\mathbb{Z}_{e}\times \prod_{i=0}^{k}{\mathbb{F}_{q_i}}$ are obtained. And it shows that all the known ZDB functions using cyclotomic cosets over $\mathbb{Z}_{n}$ are special cases of a generic construction. Moreover, applications of these ZDB functions are presented.

Sihem Mesnager, Fengrong Zhang, Chunming Tang et al.

In 2018, Pott, at al. have studied in [IEEE Transactions on Information Theory. Volume: 64, Issue: 1, 2018] the maximum number of bent components of vectorial function. They have presented serval nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from $\mathbb{F}_{2^{2k}}$ to $\mathbb{F}_{2^{2k}}$ by its univariate representation: $$ αx^{2^i}\left(x+x^{2^k}+\sum\limits_{j=1}^ργ^{(j)}x^{2^{t_j}} +\sum\limits_{j=1}^ργ^{(j)}x^{2^{t_j+k}}\right)$$ has the maximum number of {components bent functions, where $ρ\leq k$}. Further, we show that the differential spectrum of the function $ x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$ (where $i,t_1,t_2$ satisfy some conditions) is different from the binomial function $F^i(x)= x^{2^i}(x+x^{2^k})$ presented in the article of Pott et al. Finally, we provide sufficient and necessary conditions so that the functions $$Tr_1^{2k}\left(αx^{2^i}\left(Tr^{2k}_{e}(x)+\sum\limits_{j=1}^ργ^{(j)}(Tr^{2k}_{e}(x))^{2^j} \right)\right) $$ are bent.