On perfect 1-mathcalEmathcal E-error-correcting codes

We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-$mathcal{E}$-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting $q$-ary Hamming codes as subsets

Characterising bimodal collections of sets in finite groups

A collection of disjoint subsets A = {A 1 ,A 2 ,...,A m } of a finite abelian group is said to have the bimodal property if, for any non-zero group element δ, either δ never occurs as a difference between an element of A i and an element of some other set A j , or else for every element a i in A i there is an element a j ∈ A j for some j 6= i such that a i − a j = δ. This property arises in various familiar situations, such as the cosets of a fixed subgroup or in a group partition, and has applications to the construction of optimal algebraic manipulation detection (AMD) codes. In this paper, we obtain a structural characterisation for bimodal collections of sets

Some necessary conditions for vector space partitions

Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.Comment: 19 pages; corrected typos and rewritten introductio

Generalized vector space partitions

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.Comment: 12 pages, typos correcte