arXiv subjects
Publications and source records attributed to Jinquan Luo.
In this paper, new pairs of binary sequences with three cross correlation values are presented. The cross correlation values are shown to be low. Finally we present some numerical results and some open problems.
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.
There is a known best possible upper bound on the probability of undetected error for linear codes. The $[n,k;q]$ codes with probability of undetected error meeting the bound have support of size $k$ only. In this note, linear codes of full support ($=n$) are studied. A best possible upper bound on the probability of undetected error for such codes is given, and the codes with probability of undetected error meeting this bound are characterized.
Let $q=2^n$, $0\leq k\leq n-1$, $n/\gcd(n,k)$ be odd and $k\neq n/3, 2n/3$. In this paper the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(αx^{2^{2k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] is determined. As an application, the weight distribution of the binary cyclic code $\cC$, with parity-check polynomial $h_1(x)h_2(x)h_3(x)$ where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $π^{-1}$, $π^{-(2^k+1)}$ and $π^{-(2^{2k}+1)}$ respectively for a primitive element $π$ of $\bF_q$, is also determined.
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}.
Let $q=p^n$ with $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k\neq n/2$. In this paper we determine the value distribution of following exponential(character) sums \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1})}\quad(α\in \bF_{p^m},β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1}+\ga x)}\quad(α\in \bF_{p^m},β,\ga\in \bF_{q})\] 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 $d=\gcd(n,k)$, and $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $π^{-1}$, $π^{-(p^k+1)}$ and $π^{-(p^{3k}+1)}$ 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.
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}.
Let $q=2^n$, $0\leq k\leq n-1$ and $k\neq n/2$. In this paper we determine the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1})}\quad(α,β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_{2^n}\ra \bF_2$ is the canonical trace mapping. 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 $π^{-(2^{3k}+1)}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of binary m-sequences.