Optimal additive quaternary codes of dimension $3.5$
Sascha Kurz
TL;DR
The paper resolves the optimal parameters for additive quaternary codes of dimension $k=3.5$ and advances partial results for $k=4$, by linking code parameters to $(n,r,s)$-systems in $ ext{PG}(r-1,2)$ and exploiting both geometric constructions and computational bounds. A generalized Solomon–Stiffler approach demonstrates that the Griesmer bound for $n_{r/2}(s)$ is attainable for sufficiently large $s$, and a detailed combinatorial framework (including partitionable types and vector-space partitions) yields asymptotic formulas and an algorithmic path to exact values in the large-$s$ regime. The core results give explicit formulas and constructions for $n_{3.5}(s)$ (notably $n_{3.5}(31t)=127t$ and related shifts) and for $n_4(s)$ (e.g., $n_4(21t)=85t$ with several offsets), supplemented by extensive ILP-based constructions and a precise analysis of binary codes associated with the quaternary additive codes. These findings not only determine many exact values but also illuminate the interplay between additive quaternary codes, projective geometry, and binary code theory, offering concrete templates and computational benchmarks for remaining open cases.
Abstract
After the optimal parameters of additive quaternary codes of dimension $k\le 3$ have been determined there is some recent activity to settle the next case of dimension $k=3.5$. Here we complete dimension $k=3.5$ and give partial results for dimension $k=4$. We also solve the problem of the optimal parameters of additive quaternary codes of arbitrary dimension when assuming a sufficiently large minimum distance.