arXiv subjects

Yuansheng Tang

Publications and source records attributed to Yuansheng Tang.

9 recordsLinked to original sources

arXiv2026

On the degradations of Binary-Input Discrete Memoryless Channels

For the polar codes introduced by Arikan in 2009, the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding, it is crucial to evaluate the reliability of the synthetic channels resulted in the code construction. Since the synthetic channels have an output alphabet that grows exponentially with the code length, an effective method for faithfully evaluating their reliability is to replace them with degradations of manageable alphabet size. The main aim of this paper is to find the optimal degradations of symmetric BIDMCs. We determine all the degradations with the minimum probability of error decoding and give a necessary condition for the degradations with the maximum symmetric capacity. Finally, based on this necessary condition, we propose an efficient algorithm for finding degradation schemes that maximize the symmetric capacity.

cs.IT↗

arXiv2025

On the Arikan Transformations of Binary-Input Discrete Memoryless Channels

The polar codes introduced by Arikan in 2009 achieve the capacity of binary-input discrete memoryless channels (BIDMCs) with low complexity encoding and decoding. Identifying the unreliable synthetic channels, generated by Arikan transformation during the construction of these polar codes, is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no efficient and practical approach to evaluate their reliability in general. To tackle this problem, by converting the generation of synthetic channels in polar code construction into algebraic operations, in this paper we develop a method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.

cs.IT↗

arXiv2025

On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels

Polar codes introduced by Arikan in 2009 are the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding. Identifying unreliable synthetic channels in polar code construction is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no effective approach to evaluate their reliability, except in the case that the underlying channels are binary erasure channels. This paper defines equivalence and symmetry based on the likelihood ratio profile of BIDMCs and characterizes symmetric BIDMCs as random switching channels (RSCs) of binary symmetric channels. By converting the generation of synthetic channels in polar code construction into algebraic operations on underlying channels, some compact representations of RSCs for these synthetic channels are derived. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.

cs.IT↗

arXiv2022

Girth of the algebraic bipartite graph $D(k,q)$

For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph $D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is $q$-regular, edge-transitive and of girth at least $k+4$. Its exact girth $g=g(D(k,q))$ was conjectured in 1995 to be $k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $\frac{k+5}{2}|_p(q-1)$, where $p$ is the characteristic of $\mathbb{F}_q$ and $m|_pn$ means that $m$ divides $p^r n$ for some nonnegative integer $r$. In this paper, for $t\geq 1$ we prove that (a) $g(D(4t+2,q))=g(D(4t+1,q))$; (b) $g(D(4t+3,q))=4t+8$ if $g(D(2t,q))=2t+4$; (c) $g(D(8t,q))=8t+4$ if $g(D(4t-2,q))=4t+2$; (d) $g(D(2^{s+2}(2t-1)-5,q))=2^{s+2}(2t-1)$ if $p\geq 3$, $(2t-1)|_p(q-1)$ and $2^s\|(q-1)$. A simple upper bound for the girth of $D(k,q)$ is proposed in the end of this paper.

math.CO↗

arXiv2022

On the girth cycles of the bipartite graph $D(k,q)$

For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph $D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is $q$-regular, edge-transitive and of girth at least $k+4$. For its exact girth $g=g(D(k,q))$, Füredi et al. (1995) conjectured $g=k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $(k+5)/2$ is the product of an arbitrary factor of $q-1$ and an arbitrary power of the characteristic of $\mathbb{F}_q$. In this paper, we determine all the girth cycles of $D(k,q)$ for $3\leq k\leq 5$, $q>3$, and those for $3\leq k\leq8$, $q=3$.

math.CO↗

arXiv2020

On the characterization of some algebraically defined bipartite graphs of girth eight

For any field $\mathbb{F}$ and polynomials $f_{2},f_{3}\in\mathbb{F}[x,y]$, let $Γ_{\mathbb{F}}(f_{2},f_{3})$ denote the bipartite graph with vertex partition $P\cup L$, where $P$ and $L$ are two copies of $\mathbb{F}^{3}$, and $(p_{1},p_{2},p_{3})\in P$ is adjacent to $[l_{1},l_{2},l_{3}]\in L$ if and only if $p_{2}+l_{2}=f_{2}(p_{1},l_{1})$ and $p_{3}+l_{3}=f_{3}(p_{1},l_{1})$. The graph $Γ_{3}(\mathbb{F})=Γ_{\mathbb{F}}(xy,xy^{2})$ is known to be of girth eight. When $\mathbb{F}=\mathbb{F}_q$ is a finite field of odd size $q$ or $\mathbb{F}=\mathbb{F}_{\infty}$ is an algebraically closed field of characteristic zero, the graph $Γ_{3}(\mathbb{F})$ is conjectured to be the unique one with girth at least eight among those $Γ_{\mathbb{F}}(f_{2},f_{3})$ up to isomorphism. This conjecture has been confirmed for the case that both $f_{2},f_{3}$ are monomials over $\mathbb{F}_q$, and for the case that at least one of $f_{2},f_{3}$ is a monomial over $\mathbb{F}_{\infty}$. If one of $f_{2},f_{3}\in\mathbb{F}_q[x,y]$ is a monomial, it has also been proved the existence of a positive integer $M$ such that $G=Γ_{\mathbb{F}_{q^{M}}}(f_2,f_3)$ is isomorphic to $Γ_{3}(\mathbb{F}_{q^{M}})$ provided $G$ has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials $f_2,f_3$ is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials $f_2,f_3$ are characterized completely.

math.CO↗

arXiv2017

On a Generalization of the Bipartite Graph $D(k,q)$

In this paper, we deal with a generalization $Γ(Ω,q)$ of the bipartite graphs $D(k,q)$ proposed by Lazebnik and Ustimenko, where $Ω$ is a set of binary sequences that are adopted to index the entries of the vertices. A few sufficient conditions on $Ω$ for $Γ(Ω,q)$ to admit a variety of automorphisms are proposed. A sufficient condition for $Γ(Ω,q)$ to be edge-transitive is proposed further. A lower bound of the number of the connected components of $Γ(Ω,q)$ is given by showing some invariants for the components. For $Γ(Ω,q)$, paths and cycles which contain vertices of some specified form are investigated in details. Some lower bounds for the girth of $Γ(Ω,q)$ are then shown. In particular, one can give very simple conditions on the index set $Ω$ so as to assure the generalized graphs $Γ(Ω,q)$ to be a family of graphs with large girth.

math.CO↗

arXiv2009

Exponential Sums, Cyclic Codes and Sequences: the Odd Characteristic Kasami Case

Let $q=p^n$ with $n=2m$ and $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential(character) sums \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^m (αx^{p^{m}+1})+\Tra_1^n(βx^{p^k+1})}\quad(α\in \bF_{p^m},β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^m (αx^{p^{m}+1})+\Tra_1^n(βx^{p^k+1}+\ga x)}\quad(α\in \bF_{p^m},β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_q\ra \bF_p$ and $\Tra_1^m: \bF_{p^m}\ra\bF_p$ are the canonical trace mappings and $ζ_p=e^{\frac{2πi}{p}}$ is a primitive $p$-th root of unity. As applications: (1). We determine the weight distribution of the cyclic codes $\cC_1$ and $\cC_2$ over $\bF_{p^t}$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $t$ is a divisor of $d=\gcd(m,k)$, and $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $π^{-1}$, $π^{-(p^k+1)}$ and $π^{-(p^m+1)}$ over $\bF_{p^t}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences. This paper extends the results in \cite{Zen Li}.

cs.IT↗

arXiv2009

Cyclic Codes and Sequences: the Generalized Kasami Case

Let $q=2^n$ with $n=2m$ . Let $1\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^m (αx^{2^{m}+1})+\Tra_1^n(βx^{2^k+1})}\quad(α\in \bF_{2^m},β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^m (αx^{2^{m}+1})+\Tra_1^n(βx^{2^k+1}+\ga x)}\quad(α\in \bF_{2^m},β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_q\ra \bF_2$ and $\Tra_1^m: \bF_{p^m}\ra\bF_2$ are the canonical trace mappings. As applications: (1). We determine the weight distribution of the binary cyclic codes $\cC_1$ and $\cC_2$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $π^{-1}$, $π^{-(2^k+1)}$ and $π^{-(2^m+1)}$ over $\bF_{2}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences. This paper is the binary version of Luo, Tang and Wang\cite{Luo Tan} and extends the results in Kasami\cite{Kasa1}, Van der Vlugt\cite{Vand2} and Zeng, Liu and Hu\cite{Zen Liu}.

cs.IT↗