Extended cyclic codes, maximal arcs and ovoids

A family of ovoids in PG(3, 2^m) from cyclic codes

Ovoids in $\PG(3, q)$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families are known. The first is the elliptic quadratics and the second is the Tits ovoids. In this article, we present a family of ovoids in $\PG(3, 2^m)$ for all $m$ which are from a family of irreducible cyclic codes.

An algorithm for constructing some maximal arcs in $\PG(2,q^2)$

In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order $q^2$. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in $\PG(3,q)$ and, in particular, it does apply to the Suzuki--Tits ovoid. In this paper, we describe an algorithm for obtaining a possible representation of such arcs in $\PG(2,q^2)$.

PG(3,q)的循环表示中的卵形体

Ovoids in the cyclic presentation of PG(3,q)

研究PG(3,q)的循环表示中的卵形体，分析已知卵形体的结构，并通过根单位和多项式零点描述Suzuki-Tits卵形体。

Maximal arcs and extended cyclic codes

It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and acts regularly on the set of the remaining points of ${A}$. Two alternative proofs are given, one geometric proof based on Abatangelo-Larato's characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.

Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields

In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters $(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$, where $r=M(q^2-1)/2$, in the following cases: (i) $M=1$ and $q\equiv 3\,(\mod{4})$; (ii) $M=3$ and $q\equiv 7\,(\mod{24})$; and (iii) $M=7$ and $q\equiv 11,51\,(\mod{56})$. The existence of strongly regular Cayley graphs with the above para…

在PG(3,2^m)中研究正则循环(q+1)-弧：谱刚性、下降和一个MDS准则

Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion

本文研究了PG(3,2^m)中正则循环(q+1)-弧的谱刚性与下降准则，并解决了BCH族编码的MDS条件。