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Qin Yue

Publications and source records attributed to Qin Yue.

At least 19 recordsLinked to original sources
arXiv2026

Recursive Structure of Hulls of PRM Codes

For a nonnegative integer $r$ and a positive integer $v$ satisfying \[ \frac{r(q-1)}{2} 0,\\[1.2ex] 1, & r=0. \end{cases} \] For the projective Reed-Muller code $\PRM(q,m,v)$, we determine its hull dimension: \[ \dim \Hull\bigl(\PRM(q,m,v)\bigr) = \dim \PRM(q,m,v) - \sum_{i=0}^{\ell}A_{2i+\epsilon}\bigl(v-(\ell-i)(q-1)\bigr), \] where \[ \ell=\Bigl\lfloor\frac r2\Bigr\rfloor,\qquad \epsilon= \begin{cases} 0, & r\ \text{is even}, 1, & r\ \text{is odd}. \end{cases} \] This formula applies in the open lower-half range $ 0<v<\frac{m\Qm}{2}, $ equivalently for $v\in I_r$ with $m\ge r+1$; the range $ \frac{m\Qm}{2}<v<m\Qm $ is then obtained by S\o rensen's duality theorem \cite{Sorensen}.

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arXiv2024

Generator polynomials of cyclic expurgated or extended Goppa codes

Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let $\Bbb F_q$ be a finite field with $q=2^l$ elements, where $l$ is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism $A \in PGL_2(\Bbb F_q)$. Moreover, we provide some examples to support our findings.

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arXiv2024

Determining hulls of generalized Reed-Solomon codes from algebraic geometry codes

In this paper, we provide conditions that hulls of generalized Reed-Solomon (GRS) codes are also GRS codes from algebraic geometry codes. If the conditions are not satisfied, we provide a method of linear algebra to find the bases of hulls of GRS codes and give formulas to compute their dimensions. Besides, we explain that the conditions are too good to be improved by some examples. Moreover, we show self-orthogonal and self-dual GRS codes.

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arXiv2021

Extended Irreducible Binary Sextic Goppa codes

Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $Γ(L, g)$ of degree $6$ and length $2^n+1$.

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arXiv2021

Binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $\mathbb{F}_{q}$ satisfying $γg(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=\left(\begin{array}{cc} a&b\\1&d\end{array}\right)\in PGL_2(\Bbb F_{q})$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne γ\in \Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where $s$ is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.

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arXiv2021

Ternary primitive LCD BCH codes

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.

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arXiv2020

Further factorization of $x^n-1$ over finite fields (II)

Let $\Bbb F_q$ be a finite field with $q$ elements. Let $n$ be a positive integer with radical $rad(n)$, namely, the product of distinct prime divisors of $n$. If the order of $q$ modulo $rad(n)$ is either 1 or a prime, then the irreducible factorization and a counting formula of irreducible factors of $x^n-1$ over $\Bbb F_q$ were obtained by Mart\'ınez, Vergara, and Oliveira (Des Codes Cryptogr 77 (1) : 277-286, 2015) and Wu, Yue, and Fan (Finite Fields Appl 54: 197-215, 2018). In this paper, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb F_q[x]$ and calculate the number of the irreducible factors when the order of $q$ modulo $rad(n)$ is a product of two primes.

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arXiv2020

MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes

Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near MDS codes are very important in coding theory and practice. Thus, it is interesting to construct self-dual MDS or self-dual near MDS codes. In this paper, we not only give check matrices of dual codes of twisted generalized Reed-Solomon codes (TGRS codes) but also present the efficient and necessary condition of self-dual TGRS codes. Moreover, we construct several classes of self-dual MDS or self-dual near MDS codes from TGRS codes.

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arXiv2020

Optimal minimal Linear codes from posets

Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun {\em et al.} We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the condition of Ashikhmin-Barg.

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arXiv2020

Self-reciprocal and self-conjugate-reciprocal irreducible factors of $x^n-λ$ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-λ$, $λ\in \Bbb F_q^*$, over $\Bbb F_q$, which are just open questions posed by Boripan {\em et al} (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.

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arXiv2019

On the Hamming distances of repeated-root cyclic codes of length $5p^s$

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let $p$ be a prime with $p\ge 7$. We determine the weight distributions of all cyclic codes of length $5$ over $\f_q$ and the Hamming distances of all repeated-root cyclic codes of length $5p^s$ over $\F_q$, where $q=p^m$ and both $s$ and $m$ are positive integers. Furthermore, we find all MDS cyclic codes of length $5p^s$ and take quantum synchronizable codes from repeated-root cyclic codes of length $5p^s$.

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arXiv2019

New quaternary codes derived from posets of the disjoint union of two chains

Based on the generic construction of linear codes, we construct linear codes over the ring $\Bbb Z_4$ via posets of the disjoint union of two chains. We determine the Lee weight distributions of the quaternary codes. Moreover, we obtain some new linear quaternary codes and good binary codes by using the Gray map.

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arXiv2019

A class of functions with low-valued Walsh spectrum

Let $l\equiv 3\pmod 4$, $l\ne 3$, be a prime, $N=l^2$, $f=\frac{l(l-1)}2$ the multiplicative order of a prime $p$ modulo $N$, and $q=p^f$. In this paper, we investigate the Walsh spectrum of the monomial functions $f(x)={\rm Tr}_{q/p}(x^{\frac{q-1}{l^2}})$ in index two case. In special, we explicitly present the value distribution of the Walsh transform of $f(x)$ if $1+l=4p^h$, where $h$ is a class number of $\Bbb Q(\sqrt{-l})$.

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arXiv2019

Optimal few-weight codes from simplicial complexes

Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\Bbb F_2+u\Bbb F_2$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.

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arXiv2017

Further factorization of $x^n-1$ over a finite field

Let $\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive integer. Martínez, Vergara and Oliveira \cite{MVO} explicitly factorized $x^{n} - 1$ over $\Bbb F_q$ under the condition of $rad(n)|(q-1)$. In this paper, suppose that $rad(n)\nmid (q-1)$ and $rad(n)|(q^w-1)$, where $w$ is a prime, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb F_q[x]$ and count the number of its irreducible factors.

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arXiv2017

A construction of $q$-ary linear codes with two weights

Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of $q$-ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters.

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arXiv2016

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.

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