Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Cyclic codes have been an important topic of both mathematics and engineering for decades. They have been widely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic codes and their weight distributions. Information about the duals of these codes is also given when it is available.

Three-weight cyclic codesand their weight distributions

We construct two new infinite families of trace codes of dimension $2m$, over the ring $\mathbb{F}_p+u\mathbb{F}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1)^2$ and $2(p^m-1)^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\equiv 3 \pmod{4},$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3,$ or $p\ge 5$ and $m\ge 4.$ Applications to secret sharing schemes are given.

Two new families of two-weight codes

Minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two-party computation. In this paper, we focus on constructing minimal linear codes with $\frac {w_{\min }}{w_{\max }}\leq \frac {p-1}{p}$ for any odd prime $p$ based on a generic construction of linear codes, where $w_{\min }$ and $w_{\max }$ denote the minimum and maximum nonzero weights in a code, respectively. First, we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal. Second, we also present an infinite family of minimal linear codes by employing partial spreads, which can be viewed as a generalization of the construction of Ding et al. In addition, we determine the weight distributions of all these minimal linear codes.

Three Classes of Minimal Linear Codes Over the Finite Fields of Odd Characteristic

Structure and performance of generalized quasi-cyclic codes

We construct an infinite family of three-Lee-weight codes of dimension 2m, where m is singly-even, over the ring Fp+uFp$\mathbb {F}_{p}+u\mathbb {F}_{p}$ with u2=0. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain an infinite family of abelian p-ary three-weight codes. When m is odd, and pź3 (mod 4), we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. An application to secret sharing schemes is given.

Two and three weight codes over Fp+uFp$\mathbb {F}_{p}+u\mathbb {F}_{p}$

We construct an infinite family of two-Lee-weight codes over the ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$ . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. By Gray mapping, we obtain an infinite family of abelian binary two-weight codes. They are shown to be optimal by application of the Griesmer bound. An application to secret sharing schemes is given.

Optimal Two-Weight Codes From Trace Codes Over $\mathbb {F}_2+u\mathbb {F}_2$

Two-weight and three-weight codes from trace codes over Fp+uFp+vFp+uvFp

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p,$ where $u^2=0,v^2=0,uv=vu.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over $\mathbb{F}_p$. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

pdf/two-weight-and-three-weight-codes-from-trace-codes-over-amjczh484o.pdf

Two-weight and three-weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$

pdf/good-self-dual-generalized-quasi-cyclic-codes-exist-2cp13npyc5.pdf

Good self-dual generalized quasi-cyclic codes exist

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