非射影双权重码

一类投射三元双权码的enumeration及其与三次Segre簇的关系

An enumeration of certain projective ternary two-weight codes and their relationship to the cubic Segre variety

我们详细地列举了3阶域上的五维射影空间中在7阶元素下不变的所有两交集，其中包括Hill（1973）和Gulliver（1996）的例子。直到射影等价，有6635个这样的两交集。

有限域上的最大二次曲面与投影Reed-Muller码的最小码字

Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes

本文基于有限域上的最大二次曲面研究二阶投影Reed-Muller码的最小码字分类，证明了除F₂特例外，任何两个绝对不可约二次曲面若包含关系成立则必相等，并给出了最小码字的精确刻画及各重量下的数量。

Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order $p^2$

The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a m…

On projective $q^r$-divisible codes

A projective linear code over $\mathbb{F}_q$ is called $Δ$-divisible if all weights of its codewords are divisible by $Δ$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $\mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes.

Connectedness of projective codes in the Grassmann graph

Using the concept of projective systems for linear codes and elementary linear algebra, we show that projective $[n,k]_q$ codes form a connected subgraph in the Grassmann graph consisting of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field.

Linear sets on the projective line with complementary weights

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021)…

The fourth smallest Hamming weight in the code of the projective plane over $\mathbb{Z}/p \mathbb{Z}$

Let $p$ be a prime and let $C_p$ denote the $p$-ary code of the projective plane over ${\mathbb Z}/p\mathbb{Z}$. It is well known that the minimum weight of non-zero words in $C_p$ is $p+1$, and Chouinard proved that, for $p \geq 3$, the second and third minimum weights are $2p$ and $2p+1$. In 2007, Fack et. al. determined, for $p\geq 5$, all words of $C_p$ of these three weights. In this paper we recover all these results and also prove that, for $p \geq 5$, the fourth minimum weight of $C_p$ is $3p-3$. The problem of determining all words of weight $3p-3$ remains open.

A note on deformations and mutations of fake weighted projective planes

It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Mink…

Moments of isotropic measures and optimal projective codes

In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtained energy bounds are sharp for several infinite families of codes.