Partial Partitions of Vector Spaces Arising from the Construction of Byte Error Controi Codes

Abstract

We investigate collections $H=\{H_1,H_2,\dots ,H_m\}$ of pairwise disjoint $w$-subsets $H_i$ of an $r$-dimensional vector space $V$ over $GF(q)$ that arise in the construction of byte error control codes. The main problem is to maximize $m$ for fixed $w,r,$ and $q$ when $H$ is required to satisfy a subset of the following properties: (i) each $H_i$ is linearly independent; (ii) $H_i\cap H_j=\{0\}$ if $i\ne j$; (iii) $(H_i)\cap (H_j)=\{0\}$ if $i\ne j$;( iv) any two elements of $H_i\cup H_j$ are linearly independent;(v) any three elements of $H_1\cup H_2\cup \cdots \cup H_m$ are linearly independent.
Here $(x)$ denotes the subspace of $V$ spanned by $X$. Solutions to these problems yield linear block codes which are useful in controlling various combinations of byte and single bit errors in computer memories. For $r=w+1$ and for small values of $w$ the problem is solved or nearly solved. We list a variety of methods for constructing such partial partitions and give several bounds on $m$.