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

We investigate collections \( {H} = \{H_1, H_2, \ldots, H_m\}\) 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 \(i \neq j\); (iii) \((H_i) \cap (H_j) = \{0\}\) if \(i \neq j\);( 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 \(m\).