arXiv subjects

Yanfeng Qi

Publications and source records attributed to Yanfeng Qi.

At least 19 recordsLinked to original sources

arXiv2023

Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes

Sum-rank-metric codes have wide applications in the multishot network coding and the distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp type decoding algorithms were proposed and studied. They are sum-rank versions of Reed-Solomon codes and BCH codes in the Hamming metric. In this paper, we construct binary linear sum-rank-metric codes of the matrix size $2 \times 2$, from BCH, Goppa and additive quaternary Hamming metric codes. Larger sum-rank-metric codes than these sum-rank BCH codes of the same minimum sum-rank distances are obtained. Then a reduction of the decoding in the sum-rank-metric to the decoding in the Hamming metric is given. Fast decoding algorithms of BCH and Goppa type binary linear sum-rank-metric codes of the block length $t$ and the matrix size $2 \times 2$, which are better than these sum-rank BCH codes, are presented. These fast decoding algorithms for BCH and Goppa type binary linear sum-rank-metric codes of the matrix size $2 \times 2$ need at most $O(t^2)$ operations in the field ${\bf F}_4$. Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes of the matrix size $2 \times 2$ satisfying $$R_{sr}(\delta_{sr}) \geq 1-\frac{1}{2}(H_4(\frac{4}{3}\delta_{sr})+H_4(2\delta_{sr})),$$ can be constructed from Goppa codes.

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arXiv2021

An Open Problem on the Bentness of Mesnager's Functions

Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }), $$ where $m$ is an even positive integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$. We show that $ f_{a,b}$ is a bent function if the Kloosterman sum $$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*} (-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$ equals $4$, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory.

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arXiv2019

Minimal linear codes from characteristic functions

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb{F}_q$, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.

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arXiv2019

Codebooks from generalized bent $\mathbb{Z}_4$-valued quadratic forms

Codebooks with small inner-product correlation have application in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Welch bound or the Levenshtein bound. This paper presented a class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$. Cryptogr. Commun. 9(1), 41-53, 2017). By using these generalized bent $\mathbb{Z}_4$-valued quadratic forms, we constructs optimal codebooks achieving the Levenshtein bound. These codebooks have parameters $(2^{2m}+2^m,2^m)$ and alphabet size $6$.

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arXiv2019

A class of narrow-sense BCH codes over $\mathbb{F}_q$ of length $\frac{q^m-1}{2}$

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $ \frac{q^m-1}{2}$ over $\mathbb{F}_q$ with special trace representation, where $q$ is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders $δ_i$ of cyclotomic cosets modulo $ \frac{q^m-1}{2}$, we prove that narrow-sense BCH codes of length $ \frac{q^m-1}{2}$ with designed distance $δ_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2}$ have the corresponding trace representation, and have the minimal distance $d=δ_i$ and the Bose distance $d_B=δ_i$, where $1\leq i\leq \lfloor \frac{m+3}{4} \rfloor$.

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arXiv2017

New characterization and parametrization of LCD Codes

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their symplectic basis. Using such a characterization,we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results of the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

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arXiv2017

On σ-LCD codes

Linear complementary pairs (LCP) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $σ$ complementary dual ($σ$-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. As Euclidean LCD codes, $σ$-LCD codes can also be used to construct LCP of codes. We show that, for $q > 2$, all q-ary linear codes are $σ$-LCD and that, for every binary linear code $\mathcal C$, the code $\{0\}\times \mathcal C$ is $σ$-LCD. Further, we study deeply $σ$-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide characterizations of $σ$-LCD GQC codes, self-orthogonal GQC codes and self-dual GQC codes, respectively. Moreover, we provide constructions of asymptotically good $σ$-LCD GQC codes. Finally, we focus on $σ$-LCD Abelian codes and prove that all Abelian codes in a semi-simple group algebra are $σ$-LCD. The results derived in this paper extend those on the classical LCD codes and show that $σ$-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

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arXiv2017

Linear codes over Fq which are equivalent to LCD codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over $\mathbb F_{q} (q>3)$ is equivalent to an Euclidean LCD code and any linear code over $\mathbb F_{q^2} (q>2)$ is equivalent to a Hermitian LCD code. Consequently an $[n,k,d]$-linear Euclidean LCD code over $\mathbb F_q$ with $q>3$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_q$ and an $[n,k,d]$-linear Hermitian LCD code over $\mathbb F_{q^2}$ with $q>2$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_{q^2}$. Hence, when $q>3$ (resp.$q>2$) $q$-ary Euclidean (resp. $q^2$-ary Hermitian) LCD codes possess the same asymptotical bound as $q$-ary linear codes (resp. $q^2$-ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.

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arXiv2017

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD MDS codes. In particular, determining the existence of $q$-ary $[n,k]$ LCD MDS codes for various lengths $n$ and dimensions $k$ is a basic and interesting problem. In this paper, we firstly study the problem of the existence of $q$-ary $[n,k]$ LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for $q>3$ there exists a $q$-ary $[n,k]$ Euclidean LCD MDS code, where $0\le k \le n\le q+1$, or, $q=2^{m}$, $n=q+2$ and $k= 3 \text{or} q-1$. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.

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arXiv2016

Complementary Dual Algebraic Geometry Codes

Linear complementary dual (LCD) codes is a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry (AG) LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from elliptic curves. Then, some explicit LCD codes from elliptic curve are presented. MDS codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we introduce a construction mechanism for obtaining LCD codes from any algebraic curve and derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.

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arXiv2016

Quasi-Perfect Lee Codes from Quadratic Curves over Finite Fields

Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius $t=1$ or dimension $n=1, 2$. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the best alternative to the perfect codes. In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present some results. Next, we present a new constructive method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to derive two classes of $2$-quasi-perfect Lee codes are given over the space $\mathbb{Z}_p^n$ for $n=\frac{p^k+1}{2}$ $(\text{with} ~p\equiv 1, -5 \mod 12 \text{and} k \text{is any integer}, \text{or} p\equiv -1, 5 \mod 12 \text{and} k \text{is an even integer})$ and $n=\frac{p^k-1}{2}$ $(\text{with}p\equiv -1, 5 \mod 12, k \text{is an odd integer} \text{and} p^k>12)$, where $p$ is an odd prime. Our codes encompass the quasi-perfect Lee codes constructed recently by Camarero and Martínez. Furthermore, we solve a conjecture proposed by Camarero and Martínez (in "quasi-perfect Lee codes of radius $2$ and arbitrarily large dimension", IEEE Trans. Inf. Theory, vol. 62, no. 3, 2016) by proving that the related Cayley graphs are Ramanujan or almost Ramanujan. The Lee codes presented in this paper have applications to constrained and partial-response channels, in flash memories and decision diagrams.

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arXiv2015

Two infinite classes of rotation symmetric bent functions with simple representation

In the literature, few $n$-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on $\mathbb{F}_2^{n}$ of the two forms: {\rm (i)} $f(x)=\sum_{i=0}^{m-1}x_ix_{i+m} + γ(x_0+x_m,\cdots, x_{m-1}+x_{2m-1})$, {\rm (ii)} $f_t(x)= \sum_{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum_{i=0}^{m-1}x_ix_{i+m}+ γ(x_0+x_m,\cdots, x_{m-1}+x_{2m-1})$, \noindent where $n=2m$, $γ(X_0,X_1,\cdots, X_{m-1})$ is any rotation symmetric polynomial, and $m/gcd(m,t)$ is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to $m$ and the other class (ii) has algebraic degree ranging from 3 to $m$.

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arXiv2015

New infinite families of p-ary weakly regular bent functions

The characterization and construction of bent functions are challenging problems. The paper generalizes the constructions of Boolean bent functions by Mesnager \cite{M2014}, Xu et al. \cite{XCX2015} and $p$-ary bent functions by Xu et al. \cite{XC2015} to the construction of $p$-ary weakly regular bent functions and presents new infinite families of $p$-ary weakly regular bent functions from some known weakly regular bent functions (square functions, Kasami functions, and the Maiorana-McFarland class of bent functions). Further, new infinite families of $p$-ary bent idempotents are obtained.

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arXiv2015

Constructing bent functions and bent idempotents of any possible algebraic degrees

Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the degree of finite fields and have algebraic degree no more than 4. In this paper, several new infinite families of bent functions are obtained by adding the the algebraic combination of linear functions to some known bent functions and their duals are calculated. These bent functions contain some previous work on infinite families of bent functions by Mesnager \cite{M2014} and Xu et al. \cite{XCX2015}. Further, infinite families of bent idempotents of any possible algebraic degree are constructed from any quadratic bent idempotent. To our knowledge, it is the first univariate representation construction of infinite families of bent idempotents over $\mathbb{F}_{2^{2m}}$ of algebraic degree between 2 and $m$, which solves the open problem on bent idempotents proposed by Carlet \cite{C2014}. And an infinite family of anti-self-dual bent functions are obtained. The sum of three anti-self-dual bent functions in such a family is also anti-self-dual bent and belongs to this family. This solves the open problem proposed by Mesnager \cite{M2014}.

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arXiv2015

Linear codes with two or three weights from weakly regular bent functions

Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. \cite{DD2015} and Zhou et al. \cite{ZLFH2015} to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves the open problem of Ding et al. \cite{DD2015}. Further, this paper constructs new linear codes with two or three weights and presents the weight distributions of these codes. They contains some optimal codes meeting certain bound on linear codes.

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arXiv2013

Special values of Kloosterman sums and binomial bent functions

Let $p\ge 7$, $q=p^m$. $K_q(a)=\sum_{x\in \mathbb{F}_{p^m}} ζ^{\mathrm{Tr}^m_1(x^{p^m-2}+ax)}$ is the Kloosterman sum of $a$ on $\mathbb{F}_{p^m}$, where $ζ=e^{\frac{2π\sqrt{-1}}{p}}$. The value $1-\frac{2}{ζ+ζ^{-1}}$ of $K_q(a)$ and its conjugate have close relationship with a class of binomial function with Dillon exponent. This paper first presents some necessary conditions for $a$ such that $K_q(a)=1-\frac{2}{ζ+ζ^{-1}}$. Further, we prove that if $p=11$, for any $a$, $K_q(a)\neq 1-\frac{2}{ζ+ζ^{-1}}$. And for $p\ge 13$, if $a\in \mathbb{F}_{p^s}$ and $s=\mathrm{gcd}(2,m)$, $K_q(a)\neq 1-\frac{2}{ζ+ζ^{-1}}$. In application, these results explains some class of binomial regular bent functions does not exits.

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arXiv2013

Effective Construction of a Class of Bent Quadratic Boolean Functions

In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determine the bentness of these functions. We present the bentness of quadratic Boolean function for two cases: $m=2^vp^r$ and $m=2^vpq$, where $p$ and $q$ are two distinct primes. Further, we give the enumeration of quadratic bent functions for the case $m=2^vpq$.

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arXiv2013

A Note on Cyclic Codes from APN Functions

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty}$ defined by $s_t=Tr((1+α^t)^e)$, where $α$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in \cite{D}.

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