arXiv subjects

Qunying Liao

Publications and source records attributed to Qunying Liao.

At least 19 recordsLinked to original sources

arXiv2026

A class of optimal authentication codes with secrecy

In this paper, a class of linear authentication codes with secrecy, which are equipped with simple encoding rules and can be easily implemented, is constructed. By means of a special Weil sum, the maximum success probabilities of impersonation attack (denoted by $P_I$) and of substitution attack (denoted by $P_S$) for these codes are explicitly derived. It is further proven that the codes are asymptotically optimal, in the sense that both the information theoretic bound and the combinatorial bound for authentication codes are asymptotically attained by $P_I$ and $P_S$.

cs.IT↗

arXiv2026

Four classes of few-weight self-orthogonal codes and their applications for LCD codes and quantum codes

Since self-orthogonal codes, few-weight codes, linear complementary dual codes(LCD codes, for short) and quantum codes have nice applications in coding theory and cryptography, they have received continuous attention. In 2024, by introducing the notion of the augment code, Heng et al.[30] constructed several classes of few-weight self-orthogonal codes basing on defining sets, which are introduced by Ding et al.[10] in 2007. In this manuscript, for two classes of defining sets, we consider the corresponding augmented codes, construct a class of projective four-weight self-orthogonal codes and three classes of four-weight self-orthogonal codes. And for two classes of these four-weight self-orthogonal linear codes, we determine the parameters of their dual codes. As applications, we construct two classes of LCD codes and a class of quantum codes. In particular, we prove that there exists a class of these LCD codes whose dual codes are almost optimal LCD codes according to the sphere packing bound, and a class of quantum codes are AMDS according to the quantum Singleton bound.

cs.IT↗

arXiv2026

The Hermitian Hull Dimensions for a Class of (L,P)-Twisted Generalized Reed-Solomon Codes

Determining the hull of linear codes has long been an important topic in coding theory. Recently, non-generalized Reed-Solomon (in short, non-GRS) codes have attracted extensive research interest. The (L,P)-twisted generalized Reed-Solomon (in short, (L,P)-TGRS) code, which is an extension of the generalized Reed-Solomon (GRS) code, constitutes a well-studied calss of non-GRS codes.There are numerous works focusing on the Euclidean hull of (L,P)-TGRS codes, while only a few results on the Hermitian hull of (L,P)-TGRS codes. In this paper, we focus on a class of (L,P)-TGRS codes C_k(a). By taking a special class of the vector a with length i(q-1), and analyze the parity of i and the relation between i and q+1, we divide three cases to fully determine the Hermitian hull dimension of C_k(a). As an application, we construct two classes of entanglement-assisted quantum error-correcting codes.

cs.IT↗

arXiv2026

The MDS or NMDS for Modified GRS codes with flexible hull dimensions and lengths

Non-generalized Reed-Solomon (in short, non-GRS) type maximum distance separable (in short, MDS), near MDS (in short, NMDS), and linear complementary dual (in short, LCD) codes, as well as the hull of linear codes have interesting practical applications in cryptography and coding theory. In this paper, we focus on a class of non-GRS codes and its extended codes, i.e., modified generalized Reed-Solomon (MGRS) codes and extended MGRS (EMGRS) codes introduced by Wang et al. in 2026. Firstly, we prove that two classes of MGRS codes and EMGRS codes are either MDS or NMDS, derive the necessary and sufficient conditions for these codes to be NMDS, and then completely determine the weight distributions for one class of these NMDS MGRS or NMDS EMGRS codes. Secondly, we construct four classes of MGRS codes which are either Euclidean LCD codes or one-dimensional Euclidean hull codes. Thirdly, we constructively prove that there exist MGRS codes with flexible Hermitian hull dimensions and lengths. In addition, we illustrate the linearly inequivalence of NMDS MGRS codes and elliptic curve NMDS codes by Schur product. Finally, some corresponding examples are given.

cs.IT↗

arXiv2026

Construction of Minimal Ternary Linear Codes with Dimension $m+2$ Via Krawtchouk Polynomials

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for ternary linear codes with dimension $m+2$ is presented, where $m$ is an integer, and a necessary and sufficient condition for this ternary linear code to be minimal is derived. Based on this condition and Krawtchouk Polynomials, a new class of minimal ternary linear codes violating the Ashikhmin-Barg condition are obtained, and then their complete weight enumerators are determined.

cs.IT↗

arXiv2026

Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs

In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist both the Sidelnikov-Shestakov attack and the Wieschebrink attack. Constructing linear complementary dual (LCD) codes and determining the hull of linear codes have long been important topics in coding theory, as they play the crucial role in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), certain communication systems and cryptography. In this paper, by utilizing a class of non-GRS type linear codes, namely, generalized Roth-Lempel (in short, GRL) codes, we firstly construct several classes of Euclidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, generalized the main results given by Wu et al. in 2021. We also present an upper bound for the number of a class of Euclidean GRL codes with 1-dimensional hull, and then for several classes of Hermitian GRL codes, we firstly derive an upper bound for the dimension of the hull, and prove that the bound is attainable. Secondly, as an application, we obtain several families of EAQECCs. Thirdly, we prove that the GRL code is non-GRS for $k >\ell$. Finally, some corresponding examples for LCD MDS codes and LCD NMDS codes are presented.

cs.IT↗

arXiv2026

The equivalent condition for GRL codes to be MDS, AMDS or self-dual

It's well known that MDS, AMDS or self dual codes have good algebraic properties, and are applied in communication systems, data storage, quantum codes, and so on. In this paper, we focus on a class of generalized Roth-Lempel linear codes which are not not equivalent to linear codes in [21],[22] and give an equivalent condition for them or their dual to be non RS MDS, AMDS or non RS self-dual and some corresponding examples.

cs.IT↗

arXiv2026

Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions

Maximum distance separable (in short, MDS), near MDS (in short, NMDS), and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing scheme. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper presents a systematic investigation into a generalized class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes characterized by $\ell$ twists, extending the structures previously introduced by Beelen et al. and Hu et al.. We first derive the explicit parity-check matrices for these codes by analyzing the properties of symmetric polynomials. Based on this algebraic framework, we establish necessary and sufficient conditions for the self-orthogonality of the proposed codes, generalizing several recent results. Leveraging these self-orthogonal structures, we construct new families of LCD MDS codes that offer greater flexibility in code length compared to existing literature. Furthermore, we provide a characterization of the NMDS property for these codes, offering a partial solution to the open problem concerning general $(\mathcal{L}, \mathcal{P})$-TGRS codes posed by Hu et al. (2025). Finally, we rigorously prove that these codes are of non-GRS type when $2k > n$, providing an improvement over previous bounds. Theoretical constructions are validated through numerical examples.

cs.IT↗

arXiv2025

On the Hamming Weight Functions of Linear Codes

Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight functions. Specifically, we develop a general framework that constructs new linear codes from the set of codewords in a given code having a fixed Hamming weight. We analyze the dimension, number of weights, and weight distribution of the constructed codes, and establish connections with the extendability of the original codes as well as the partial weight distribution of the derived codes. As a new tool, this framework enables us to establish an upper bound on the minimum weight of two-weight codes and to characterize all two-weight codes attaining this bound. Moreover, several divisibility properties concerning the parameters of two-weight codes are derived. The proposed method not only generates new families of linear codes but also provides a powerful approach for exploring the intrinsic combinatorial and geometric structures of existing codes.

cs.IT↗

arXiv2025

On the Ding and Helleseth's 9th open problem about optimal ternary cyclic codes

The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd and 6th problems were completely solved, and the 3rd, 7th, 8th and 9th problems were partially solved. In this manuscript, we focus on the 9th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer for the 9th problem, and then we construct two classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound basing on some special polynomials over finite fields

cs.IT↗

arXiv2025

Several classes of three-weight or four-weight linear codes

In this manuscript, we construct a class of projective three-weight linear codes and two classes of projective four-weight linear codes over F2 from the defining sets construction, and determine their weight distributions by using additive characters. Especially, the projective three-weight linear code and one class of projective four-weight linear codes (Theorem 4.1) can be applied in secret sharing schemes.

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arXiv2025

The asymptotic estimation for two classes of generalized Fibonacci sub-sequences

Since the $\mathrm{Fibonacci}$ sequence has good properties, it's important in theory and applications, such as in combinatorics, cryptography, and so on. In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we respectively give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{W_{mk+l}^d}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{W_{mk+l}^d}\right)^{-1}$, which generalize the asymptotic estimation results of Yuan et al. \cite{A14} in 2025.

math.CO↗

arXiv2025

The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences

In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\leq i\leq t \right)$, Theorem \ref{theorem 3.1} is just Theorems 2.1, 2.5-2.6 in \cite{A22} given by Yuan et al.

math.NT↗

arXiv2025

Four classes of LCD codes from (*)-(L,P)-twisted generalized Reed-Solomon codes

It's well-known that maximum distance separable codes (in short, MDS) and linear complementary dual (in short, LCD) codes are very important in coding theory and practice. In 2023, Yue et al. [25] constructed three classes of LCD MDS codes via (*)-TGRS codes. Recently, Wu et al. [27] generalized the results given by Yue et al. and constructed several classes of LCD MDS codes. In this paper, we unify their constructions by defining the (*)-(L,P)-twisted generalized Reed-Solomon (in short, (*)-(L,P)-TGRS) code, give the parity-check matrix of (*)-(L,P)-TGRS codes, and then construct four classes of LCD codes. Finally, some corresponding examples are given.

cs.IT↗

arXiv2025

The extended code for a class of generalized Roth-Lempel codes and their properties

As we all know, many interesting and important codes are obtained by modifying or combining existing codes. In this paper, we focus on generalized Roth-Lempel (in short, GRL) codes and define a class of extended codes, i.e., the extended generalized Roth-Lempel (in short, EGRL) code. And then for a special class of EGRL codes, we give a parity-check matrix and establish a necessary and sufficient condition for the EGRL code or its dual code to be MDS or AMDS, respectively. Finally, we construct a class of NMDS EGRL codes which is the generalization of the constructions given by Han et al. in 2023, and then completely determine its weight distribution.

cs.IT↗

arXiv2025

Maximum Likelihood Estimation Based Complex-Valued Robust Chinese Remainder Theorem and Its Fast Algorithm

Recently, a multi-channel self-reset analog-to-digital converter (ADC) system with complex-valued moduli has been proposed. This system enables the recovery of high dynamic range complex-valued bandlimited signals at low sampling rates via the Chinese remainder theorem (CRT). In this paper, we investigate complex-valued CRT (C-CRT) with erroneous remainders, where the errors follow wrapped complex Gaussian distributions. Based on the existing real-valued CRT utilizing maximum likelihood estimation (MLE), we propose a fast MLE-based C-CRT (MLE C-CRT). The proposed algorithm requires only $2L$ searches to obtain the optimal estimate of the common remainder, where $L$ is the number of moduli. Once the common remainder is estimated, the complex number can be determined using the C-CRT. Furthermore, we obtain a necessary and sufficient condition for the fast MLE C-CRT to achieve robust estimation. Finally, we apply the proposed algorithm to ADCs. The results demonstrate that the proposed algorithm outperforms the existing methods.

eess.SP↗

arXiv2025

Two classes of NMDS codes from Roth-Lempel codes

Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's and Zheng's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.

cs.IT↗

arXiv2025

On the Ding and Helleseth's 8th open problem about optimal ternary cyclic codes

The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems due to the efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd, 6th and 7th problems were completely solved, the 3rd, 8th and 9th problems were incompletely solved. In this manuscript, we focus on the 8th problem. By determining the root set of some special polynomials over finite fields, we present a counterexample and a sufficient condition for the ternary cyclic code $\mathcal{C}_{(1, e)}$ optimal. Furthermore, basing on the properties of finite fields, we construct a class of optimal ternary cyclic codes with respect to the Sphere Packing Bound, and show that these codes are not equivalent to any known codes.

cs.IT↗