Two theorems on perfect codes

Wetensch. publicati

An Overview of the Different Kinds of Vector Space Partitions

In a finite vector space V (n,q), where V is n-dimensional over a finite field with q elements, a collection P of subspaces is called a vector space partition. The property of this set P is that any vector that is not zero may be found in exactly one element of P. Partitions of vector spaces have strong ties to design theory, error-correcting algorithms, and finite projective planes. The first portion of my talk will focus on the mathematical fields that share common ground with vector space partitions. The rest of the lecture will go over some of the most well-known results on vector space partition classification. Heden and Lehmann's result on vector space partitions and maximal partial spreads, as well as El-Zanati et al.'s recent findings on the types found in spaces V(n, 2) for n = 8 or less, the Beutelspacher and Heden theorem on T-partitions, and their newly established condition for the existence of a vector space partition will all be covered. Furthermore, I will demonstrate Heden's theorem about the tail length of a vector space split. Finally, I shall provide some historical notes

Partitions of codes

In this thesis we look at coding theory wherein we introduce the concept of perspective, a generalisation on the minimum distance of a code, which naturally leads to a partition of the code. Subsequently we introduce focused splittings, which shall be shown to be a generalisation of perfect codes. We investigate the existence of such objects, and address questions such as the complexity of finding a focused splittings, which we show to be NPComplete. We analyse the symmetries of focused splittings. We use focused splittings to address the problem of error correction and we construct an encoding method based on them. Finally we test this construction for various classes of focused splittings