The Gaussian normal basis and its trace basis over finite fields

AbstractIt is well known that normal bases are useful for implementations of finite fields in various applications including coding theory, cryptography, signal processing, and so on. In particular, optimal normal bases are desirable. When no optimal normal basis exists, it is useful to have normal bases with low complexity. In this paper, we study the type k(⩾1) Gaussian normal basis N of the finite field extension Fqn/Fq, which is a classical normal basis with low complexity. By studying the multiplication table of N, we obtain the dual basis of N and the trace basis of N via arbitrary medium subfields Fqm/Fq with m|n and 1⩽m⩽n. And then we determine all self-dual Gaussian normal bases. As an application, we obtain the precise multiplication table and the complexity of the type 2 Gaussian normal basis and then determine all optimal type 2 Gaussian normal bases

Several classes of projective few-weight linear codes and their applications

It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces, strongly regular graphs and combinatorial designs \cite{RC1986,CD2018,P1972}. Here, we present the following two applications.Comment: arXiv admin note: text overlap with arXiv:2107.0244

The (+)(+)-extended twisted generalized Reed-Solomon code

In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the $(+)$-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal, and then construct several classes of self-dual $(+)$-TGRS codes and almost self-dual $(+)$-ETGRS codes

The bb-weight distribution for MDS codes

For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, a $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution for $\mathcal{C}$. In this paper, by calculating the number of codewords in $\mathcal{C}$ with special shape, we obtain the $b$-weight distribution for $\mathcal{C}$, and then generalize Theorem $1$ in \cite{ML}

On the divisibility of power LCM matrices by power GCD matrices

summary:Let $S=\lbrace x_1,\dots ,x_n\rbrace$ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $(x_i,x_j)$ (resp. the $e$-th power of the least common multiple $[x_i,x_j]$) as the $(i,j)$-entry of the matrix by $((x_i, x_j)^e)$ (resp. $([x_i, x_j]^e))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i, x_j]\in S$) for all $1\le i,j \le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer $e\ge 1$, the $n\times n$ power GCD matrix $((x_i, x_j)^e)$ defined on an odd-gcd-closed (resp. odd-lcm-closed) set $S$ divides the $n\times n$ power LCM matrix $([x_i, x_j]^e)$ defined on $S$ in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for $n\le 3$ but they are both not true for $n\ge 4$