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Qunying Liao

Publications and source records attributed to Qunying Liao.

At least 37 records · Page 2Linked to original sources
arXiv2025

The error-correcting pair for several classes of NMDS linear codes

The error-correcting pair is a general algebraic decoding method for linear codes. The near maximal distance separable (NMDS) linear code is a subclass of linear codes and has applications in secret sharing scheme and communication systems due to the efficient performance, thus we focus on the error-correcting pair of NMDS linear codes. In 2023, He and Liao showed that for an NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+1$ or $2\ell+2$, if $\mathcal{C}$ has an $\ell$-error-correcting pair $\left( \mathcal{A}, \mathcal{B} \right)$, then the parameters of $\mathcal{A}$ have 6 or 10 possibilities, respectively. In this manuscript, basing on Product Singleton Bound, we give several necessary conditions for that the NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+1$ has an $\ell$-error-correcting pair $(\mathcal{A}, \mathcal{B})$, where the parameters of $\mathcal{A}$ is the 1st, 2nd, 4th or 5th case, then basing on twisted generalized Reed-Solomon codes, we give an example for that the parameters of $\mathcal{A}$ is the 1st case. Moreover, we also give several necessary conditions for that the NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+2$ has an $\ell$-error-correcting pair $(\mathcal{A}, \mathcal{B})$, where the parameters of $\mathcal{A}$ is the 2nd, 4th, 7th or 8th case, then we give an example for that the parameters of $\mathcal{A}$ is the 1st or 2nd case, respectively.

cs.IT↗
arXiv2023

Infinite families $2$-designs from binary projective three-weight codes

Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary three-weight code $\mathcal{C}$ with length $n$, let $A_{n}(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ with Hamming weight $n$, then we show that $\mathcal{C}$ holds $2$-designs when $\mathcal{C}$ is projective and $A_{n}(\mathcal{C})=1$. Furthermore, by extending some certain binary projective two-weight codes and basing on the defining set method, we construct two classes of binary projective three-weight codes which are suitable for holding $2$-designs.

cs.IT↗
arXiv2023

The $[1,0]$-twisted generalized Reed-Solomon code

In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the $[1,0]$-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $[1,0]$-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual $[1,0]$-TGRS codes. Finally, basing on these self-dual or almost self-dual $[1,0]$-TGRS codes, we obtain some LCD $[1,0]$-TGRS codes.

cs.IT↗
arXiv2022

The $(+)$-extended twisted generalized Reed-Solomon code

In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the $(+)$-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal, and then construct several classes of self-dual $(+)$-TGRS codes and almost self-dual $(+)$-ETGRS codes.

cs.IT↗
arXiv2022

Several classes of projective few-weight linear codes and their applications

It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces, strongly regular graphs and combinatorial designs \cite{RC1986,CD2018,P1972}. Here, we present the following two applications.

cs.IT↗
arXiv2022

Self-dual twisted generalized Reed-Solomon codes

In this paper, by using some properties for linear algebra methods, the parity-check matrices for twisted generalized Reed-Solomon codes with any given hook $h$ and twist $t$ are presented, and then a sufficient and necessary condition for the twisted generalized Reed-Solomon code with $h\ge t$ to be self-dual is given. Furthermore, several classes of self-dual codes with small Singleton defect are constructed based on twisted generalized Reed-Solomon codes, especially some of these self-dual codes are MDS or NMDS.

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arXiv2022

The $b$-weight distribution for MDS codes

For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution of $\mathcal{C}$. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $\mathcal{C}$, we give the connection between the $b$-weight distribution and the number of codewords in shortened codes of $\mathcal{C}$ with special shape. Furthermore, note that shortened codes of $\mathcal{C}$ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method. From the above calculation, the $b$-weight distribution of $\mathcal{C}$ is determined. Our result generalies the corresonding result of Ma and Luo.

cs.IT↗
arXiv2022

Constructions For Several Few-weight Linear Codes And Their Applications

In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.

cs.IT↗
arXiv2022

A recursion formula for the generalized Euler function $ φ_e(n) $

In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the generalized Euler function $ φ_e(n) $, which is determined by some matrices related to a congruence equation modulo $ e $. Furthermore, through the recursion formula, we get the explicit formula for $ φ_5(n) $. Our results generalize the corresponding results in \cite{A4,A8,A10,A11}.

math.NT↗
arXiv2022

A new class of MDS symbol-pair codes

The symbol-pair code is a new coding framework proposed to guard against pair-errors in symbol-pair read channels. Especially, a symbol-pair code with the parameters achieving the Singleton-type bound is called an MDS symbol-pair code. In this paper, inspiring by the classical construction for Reed-Solomon codes, for any $3\le k<m\le q-2$ and $m_1=\Big\lfloor{\tiny\frac{m}{\lfloor\frac{k-1}{2}\rfloor}}\Big\rfloor$, we construct a class of $q$-ary MDS symbol-pair codes with dimension $k$ and length $n$ $(n=m+m_1, m+m_1-1)$, where $q$ is a prime power. Furthermore, for $k\in\{3,4\}$, the symbol-pair weight distributions for these codes are determined by enumerating the number of polynomials with given roots.

cs.IT↗
arXiv2022

The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is non-GRS when the code rate is less than one half. In this letter, basing on the dual code of the TGRS code or the extended TGRS code, by using the Schur product, we prove that almost all of TGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.

cs.IT↗
arXiv2022

A New Constructions of Minimal Binary Linear Codes

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 binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.

cs.IT↗
arXiv2022

Self-orthogonal generalized twisted Reed-Solomon codes

In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon code with $h+t\le k-1$ to be self-orthogonal, where $k$ is dimension, $h$ is hook and $t$ is twist. And then, we show that there is no self-orthogonal generalized twisted Reed-Solomon code under some conditions. Furthermore, several classes of self-orthogonal generalized twisted Reed-Solomon codes are constructed, and some of these codes are non-GRS self-orthogonal MDS codes or NMDS codes.

cs.IT↗
arXiv2021

The complete weight enumerator of the Reed-Solomon code with dimension two or three

It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding results in \cite{BK91,K04}.

cs.IT↗
arXiv2021

Two new classes of projective two-weight linear codes

In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight distributions are determined by employing character sums. These codes can be suitable for applications in secret sharing schemes. Furthermore, two new classes of projective two-weight codes are obtained, and then two new classes of strongly regular graphs are given.

cs.IT↗
arXiv2021

Complete weight enumerators for several classes of two-weight and three-weight linear codes

In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in \cite{CL2016, GJ2019}.

cs.IT↗
arXiv2020

Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and their asymptotic performance, arXiv:1911.11738, 2019] about a lower bound of the minimum distance of a minimal code. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.

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