arXiv subjects

Haibin Kan

Publications and source records attributed to Haibin Kan.

35 records · Page 2Linked to original sources

arXiv2022

Minimal Binary Linear Codes from Vectorial Boolean Functions

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, we obtain a necessary and sufficient condition for another generic class of linear codes from vectorial Boolean functions to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

cs.IT↗

arXiv2022

Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

Let $\mathbf{H}$ be the Cartesian product of a family of finite abelian groups. Via a polynomial approach, we give sufficient conditions for a partition of $\mathbf{H}$ induced by weighted poset metric to be reflexive, which also become necessary for some special cases. Moreover, by examining the roots of the Krawtchouk polynomials, we establish non-reflexive partitions of $\mathbf{H}$ induced by combinatorial metric. When $\mathbf{H}$ is a vector space over a finite field $\mathbb{F}$, we consider the property of admitting MacWilliams identity (PAMI) and the MacWilliams extension property (MEP) for partitions of $\mathbf{H}$. With some invariance assumptions, we show that two partitions of $\mathbf{H}$ admit MacWilliams identity if and only if they are mutually dual and reflexive, and any partition of $\mathbf{H}$ satisfying the MEP is in fact an orbit partition induced by some subgroup of $\Aut_{\mathbb{F}}(\mathbf{H})$, which is necessarily reflexive. As an application of the aforementioned results, we establish partitions of $\mathbf{H}$ induced by combinatorial metric that do not satisfy the MEP, which further enable us to provide counter-examples to a conjecture proposed by Pinheiro, Machado and Firer in \cite{39}.

cs.IT↗

arXiv2022

Isometries and MacWilliams Extension Property for Weighted Poset Metric

Let $\mathbf{H}$ be the cartesian product of a family of left modules over a ring $S$, indexed by a finite set $Ω$. We are concerned with the $(\mathbf{P},ω)$-weight on $\mathbf{H}$, where $\mathbf{P}=(Ω,\preccurlyeq_{\mathbf{P}})$ is a poset and $ω:Ω\longrightarrow\mathbb{R}^{+}$ is a weight function. We characterize the group of $(\mathbf{P},ω)$-weight isometries of $\mathbf{H}$, and give a canonical decomposition for semi-simple subcodes of $\mathbf{H}$ when $\mathbf{P}$ is hierarchical. We then study the MacWilliams extension property (MEP) for $(\mathbf{P},ω)$-weight. We show that the MEP implies the unique decomposition property (UDP) of $(\mathbf{P},ω)$, which further implies that $\mathbf{P}$ is hierarchical if $ω$ is identically $1$. For the case that either $\mathbf{P}$ is hierarchical or $ω$ is identically $1$, we show that the MEP for $(\mathbf{P},ω)$-weight can be characterized in terms of the MEP for Hamming weight, and give necessary and sufficient conditions for $\mathbf{H}$ to satisfy the MEP for $(\mathbf{P},ω)$-weight when $S$ is an Artinian simple ring (either finite or infinite). When $S$ is a finite field, in the context of $(\mathbf{P},ω)$-weight, we compare the MEP with other coding theoretic properties including the MacWilliams identity, Fourier-reflexivity of partitions and the UDP, and show that the MEP is strictly stronger than all the rest among them.

cs.IT↗

arXiv2021

A Galois Connection Approach to Wei-Type Duality Theorems

In $1991$, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei's duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei's duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result in this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb{Z}$, from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$-demimatroids defined over finite sets and $w$-demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

cs.IT↗

arXiv2021

Fourier-Reflexive Partitions Induced by Poset Metric

Let $\mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $Ω$. A given poset (i.e., partially ordered set) $\mathbf{P}=(Ω,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset metric on $\mathbf{H}$, which further leads to a partition $\mathcal{Q}(\mathbf{H},\mathbf{P})$ of $\mathbf{H}$. We prove that if $\mathcal{Q}(\mathbf{H},\mathbf{P})$ is Fourier-reflexive, then its dual partition $Λ$ coincides with the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover, $\mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in \cite{4}. We also show that with some other assumptions, $Λ$ is finer than the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$. In addition, we give some necessary and sufficient conditions for $\mathbf{P}$ to be hierarchical, and for the case that $\mathbf{P}$ is hierarchical, we give an explicit criterion for determining whether two codewords in $\hat{\mathbf{H}}$ belong to the same block of $Λ$. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

cs.IT↗

arXiv2021

Constructing new APN functions through relative trace functions

In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the form $x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,ω, ω^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,ω, ω^2,1)$ in which $ω\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking $a=ω$ and $b=c=ω^2$, we observe that such kind of quadrinomials can be rewritten as $a {\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(cx^{2^s+1})$, where $q=2^m$ and $ {\rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^q{\rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a \in \mathbb{F}_{2^n} $ such that $ a+a^q\neq 0$, and both $F$ and $G$ are quadratic functions over $\mathbb{F}_{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $ n=2m $ with $m$ being an odd positive integer: $ f(x)=a{\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(b^3x^9) $, where $ a\in \mathbb{F}_{2^n}$ such that $ a+a^q\neq 0 $ and $ b $ is a non-cube in $ \mathbb{F}_{2^n} $.

cs.IT↗

arXiv2021

Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In generally, they are simultaneously sparse, scale-free, small-world, and loopy. In this paper, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherence $H_{\rm SO}$ characterized in terms of the $\mathcal{H}_2$-norm of the vertex systems, which measures the mean deviation of vertex states from their average value. We first study numerically the coherence of some representative real-world networks. We find that their coherence $H_{\rm SO}$ scales sublinearly with the vertex number $N$. We then study analytically $H_{\rm SO}$ for a class of iteratively growing networks -- pseudofractal scale-free webs (PSFWs), and obtain an exact solution to $H_{\rm SO}$, which also increases sublinearly in $N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study $H_{\rm SO}$ for Sierpinśki gaskets, for which $H_{\rm SO}$ grows superlinearly in $N$, with a power exponent much larger than 1. Sierpinśki gaskets have the same number of vertices and edges as the PSFWs, but do not display the scale-free and small-world properties. We thus conclude that the scale-free and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of $H_{\rm SO}$.

math.NA↗

arXiv2019

On constructions and properties of $(n,m)$-functions with maximal number of bent components

For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is at most $k$. It is easily seen that all $(n,m)$-functions of the form $G(x)=(F(x),0)$ with $F(x)$ being any vectorial bent $(n,k)$-function, have the maximum number of bent components. Those simple functions $G$ are called trivial in this paper. We show that for a power $(n,n)$-function, it has such large number of bent components if and only if it is trivial under a mild condition. We also consider the $(n,n)$-function of the form $F^{i}(x)=x^{2^{i}}h({\rm Tr}^{n}_{e}(x))$, where $h: \mathbb{F}_{2^{e}} \rightarrow \mathbb{F}_{2^{e}}$, and show that $F^{i}$ has such large number if and only if $e=k$, and $h$ is a permutation over $\mathbb{F}_{2^{k}}$. It proves that all the previously known nontrivial such functions are subclasses of the functions $F^{i}$. Based on the Maiorana-McFarland class, we present constructions of large numbers of $(n,m)$-functions with maximal number of bent components for any integer $m$ in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is found that our constructions can also provide new plateaued vectorial functions.

cs.IT↗

arXiv2019

Constructing vectorial bent functions via second-order derivatives

Let $n$ be an even positive integer, and $m<n$ be one of its positive divisors. In this paper, inspired by a nice work of Tang et al. on constructing large classes of bent functions from known bent functions [27, IEEE TIT, 63(10): 6149-6157, 2017], we consider the construction of vectorial bent and vectorial plateaued $(n,m)$-functions of the form $H(x)=G(x)+g(x)$, where $G(x)$ is a vectorial bent $(n,m)$-function, and $g(x)$ is a Boolean function over $\mathbb{F}_{2^{n}}$. We find an efficient generic method to construct vectorial bent and vectorial plateaued functions of this form by establishing a link between the condition on the second-order derivatives and the key condition given by [27]. This allows us to provide (at least) three new infinite families of vectorial bent functions with high algebraic degrees. New vectorial plateaued $(n,m+t)$-functions are also obtained ($t\geq 0$ depending on $n$ can be taken as a very large number), two classes of which have the maximal number of bent components.

cs.IT↗

arXiv2017

A Novel Construction of Low-Complexity MDS Codes with Optimal Repair Capability for Distributed Storage Systems

Maximum-distance-separable (MDS) codes are a class of erasure codes that are widely adopted to enhance the reliability of distributed storage systems (DSS). In (n, k) MDS coded DSS, the original data are stored into n distributed nodes in an efficient manner such that each storage node only contains a small amount (i.e., 1/k) of the data and a data collector connected to any k nodes can retrieve the entire data. On the other hand, a node failure can be repaired (i.e., stored data at the failed node can be successfully recovered) by downloading data segments from other surviving nodes. In this paper, we develop a new approach to construction of simple (5, 3) MDS codes. With judiciously block-designed generator matrices, we show that the proposed MDS codes have a minimum stripe size α = 2 and can be constructed over a small (Galois) finite field F4 of only four elements, both facilitating low-complexity computations and implementations for data storage, retrieval and repair. In addition, with the proposed MDS codes, any single node failure can be repaired through interference alignment technique with a minimum data amount downloaded from the surviving nodes; i.e., the proposed codes ensure optimal exact-repair of any single node failure using the minimum bandwidth. The low-complexity and all-node-optimal-repair properties of the proposed MDS codes make them readily deployed for practical DSS.

cs.IT↗

arXiv2014

On the Minimum Decoding Delay of Balanced Complex Orthogonal Design

Complex orthogonal design (COD) with parameter $[p, n, k]$ is a combinatorial design used in space-time block codes (STBCs). For STBC, $n$ is the number of antennas, $k/p$ is the rate, and $p$ is the decoding delay. A class of rate $1/2$ COD called balanced complex orthogonal design (BCOD) has been proposed by Adams et al., and they constructed BCODs with rate $k/p = 1/2$ and decoding delay $p = 2^m$ for $n=2m$. Furthermore, they prove that the constructions have optimal decoding delay when $m$ is congruent to $1$, $2$, or $3$ module $4$. They conjecture that for the case $m \equiv 0 \pmod 4$, $2^m$ is also a lower bound of $p$. In this paper, we prove this conjecture.

cs.IT↗

arXiv2012

A Note on "On the Construction of Boolean Functions with Optimal Algebraic Immunity"

In this note, we go further on the "basis exchange" idea presented in \cite{LiNa1} by using Mobious inversion. We show that the matrix $S_1(f)S_0(f)^{-1}$ has a nice form when $f$ is chosen to be the majority function, where $S_1(f)$ is the matrix with row vectors $\upsilon_k(α)$ for all $α\in 1_f$ and $S_0(f)=S_1(f\oplus1)$. And an exact counting for Boolean functions with maximum algebraic immunity by exchanging one point in on-set with one point in off-set of the majority function is given. Furthermore, we present a necessary condition according to weight distribution for Boolean functions to achieve algebraic immunity not less than a given number.

cs.CR↗

arXiv2012

Complex Orthogonal Designs with Forbidden $2 \times 2$ Submatrices

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such that $\mathcal{O}^H_z \mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$. Define $\mathcal{O}_z$ a first type COD if and only if $\mathcal{O}_z$ does not contain submatrix {\pm z_j & 0; \ 0 & \pm z^*_j}$ or ${\pm z^*_j & 0; \ 0 & \pm z_j}$. It is already known that, all CODs with maximal rate, i.e., maximal $k/p$, are of the first type. In this paper, we determine all achievable parameters $[p, n, k]$ of first type COD, as well as all their possible structures. The existence of parameters is proved by explicit-form constructions. New CODs with parameters $[p,n,k]=[\binom{n}{w-1}+\binom{n}{w+1}, n, \binom{n}{w}], $ for $0 \le w \le n$, are constructed, which demonstrate the possibility of sacrificing code rate to reduce decoding delay. It's worth mentioning that all maximal rate, minimal delay CODs are contained in our constructions, and their uniqueness under equivalence operation is proved.

cs.IT↗

arXiv2012

On $2k$-Variable Symmetric Boolean Functions with Maximum Algebraic Immunity $k$

Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean function with maximum algebraic immunity $\frac{n}{2}$ is determined by the binary expansion of $n$. Based on the foregoing, all $n$-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is $(2\wt(n)+1)2^{\lfloor \log_2 n \rfloor}$

cs.CR↗

arXiv2012

Capacity Factors of a Point-to-point Network

In this paper, we investigate some properties on capacity factors, which were proposed to investigate the link failure problem from network coding. A capacity factor (CF) of a network is an edge set, deleting which will cause the maximum flow to decrease while deleting any proper subset will not. Generally, a $k$-CF is a minimal (not minimum) edge set which will cause the network maximum flow decrease by $k$. Under point to point acyclic scenario, we characterize all the edges which are contained in some CF, and propose an efficient algorithm to classify. And we show that all edges on some $s$-$t$ path in an acyclic point-to-point acyclic network are contained in some 2-CF. We also study some other properties of CF of point to point network, and a simple relationship with CF in multicast network. On the other hand, some computational hardness results relating to capacity factors are obtained. We prove that deciding whether there is a capacity factor of a cyclic network with size not less a given number is NP-complete, and the time complexity of calculating the capacity rank is lowered bounded by solving the maximal flow. Besides that, we propose the analogous definition of CF on vertices and show it captures edge capacity factors as a special case.

cs.IT↗

arXiv2011

Constructing and Counting Even-Variable Symmetric Boolean Functions with Algebraic Immunity not Less Than $d$

In this paper, we explicitly construct a large class of symmetric Boolean functions on $2k$ variables with algebraic immunity not less than $d$, where integer $k$ is given arbitrarily and $d$ is a given suffix of $k$ in binary representation. If let $d = k$, our constructed functions achieve the maximum algebraic immunity. Remarkably, $2^{\lfloor \log_2{k} \rfloor + 2}$ symmetric Boolean functions on $2k$ variables with maximum algebraic immunity are constructed, which is much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than $d$ is derived, which is $2^{\lfloor \log_2{d} \rfloor + 2(k-d+1)}$. As far as we know, this is the first lower bound of this kind.

cs.CR↗

arXiv2011

On the nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, complex orthogonal design

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p\times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such that $\mathcal{O}^H_z \mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$. Adams et al. in "The final case of the decoding delay problem for maximum rate complex orthogonal designs," IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 103-122, Jan. 2010, first proved the nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, COD. Combining with the previous result that decoding delay should be an integer multiple of $\binom{2m}{m-1}$, they solved the final case $n \equiv 2 \pmod 4$ of the decoding delay problem for maximum rate complex orthogonal designs. In this paper, we give another proof of the nonexistence of COD with parameter $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd. Our new proof is based on the uniqueness of $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ under equivalence operation, where an explicit-form representation is proposed to help the proof. Then, by proving it's impossible to add an extra orthogonal column on COD $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ when $m$ is odd, we complete the proof of the nonexistence of COD $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$.

cs.IT↗