There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Distance-Regular Graphs

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Introduction to finite fields and their applications: Factorization of Polynomials

A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.

Linear Codes from Some 2-Designs

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$\mathbb {F}_{p}+u\mathbb {F}_{p}$

In this paper, we investigate the structure and properties of cyclic codes over the ring F 2+vF 2 . We first study the relationship between cyclic codes over F 2+vF 2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length n over F 2+vF 2.

Some Results on Cyclic Codes Over ${F}_{2}+v{F}_{2}$

We construct two new infinite families of trace codes of dimension $2m$ , over the ring $\mathbb {F}_{p}+u\mathbb {F}_{p}$ , with $u^{2}=u$ , 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

Constructions of optimal LCD codes over large finite fields

We first define a new Gray map from $R=\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}^{2}_{4}$, where $u^2=1$ and study $(1+2u)$-constacyclic codes over $R$. Also of interest are some properties of $(1+2u)$-constacyclic codes over $R$. Considering their $\mathbb{Z}_4$ images, we prove that the Gray images of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic $R$-codes published by Ozen et al. in Finite Fields and Their Applications, {\bf 38}, (2016) 27-39.

Minjia Shi

Papers

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

Some Results on Cyclic Codes Over ${F}_{2}+v{F}_{2}$

Two New Families of Two-Weight Codes

Constructions of optimal LCD codes over large finite fields

On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images