Additive quaternary codes related to exceptional linear quaternary codes

The nonexistence of an additive quaternary [15,5,9]-code

We show that no additive [15,5,9]_4-code exists. As a consequence the largest dimension k such that an additive quaternary [15,k,9]_4-code exists is k=4.5.

Optimal additive quaternary codes of low dimension

An additive quaternary $[n,k,d]$-code (length $n,$ quaternary dimension $k,$ minimum distance $d$) is a $2k$-dimensional F_2-vector space of $n$-tuples with entries in $Z_2\times Z_2$ (the $2$-dimensional vector space over F_2) with minimum Hamming distance $d.$ We determine the optimal parameters of additive quaternary codes of dimension $k\leq 3.$ The most challenging case is dimension $k=2.5.$ We prove that an additive quaternary $[n,2.5,d]$-code where $d<n-1$ exists if and only if $3(n-d)\geq \lceil d/2\rceil +\lceil d/4\rceil +\lceil d/8\rceil$. In particular we construct new optimal $2.5…

四元加性码的一个渐近性质

An asymptotic property of quaternary additive codes

本文确定了四元加性码长度函数 $n_k(s)/s$ 的上极限及其达到极限的最小 $s$ 值，构造了一族满足 Griesmer 界的等重码，并利用射影几何中的多重集给出证明。

维度3.5和4的最优加法四元码

Optimal additive quaternary codes of dimension $3.5$ and $4$

研究解决维度3.5和4的加法四元码最优参数问题，并探讨大最小距离下的任意维度最优参数情况。

某些扩展二次剩余码中的特殊设计

Exceptional designs in some extended quadratic residue codes

本文证明了长度为14的扩展三元二次剩余码和长度为18的扩展四元二次剩余码中存在3-设计。

Relations between exceptional sets for additive problems

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N, that fail to have a representation as the sum of six cubes of natural numbers, is O(N^{3/7}).