  In this paper we investigate some properties of ideals in group algebras of
finite groups over fields. First, we highlight an important link between their
dimension, their minimal Hamming distance and the group order. This is a
generalized version of an uncertainty principle shown in 1992 by Meshulam.
Secondly, we introduce the notion of the Schur product of ideals in group
algebras and investigate the module structure and the dimension of the Schur
square. We give a structural result on ideals that coincide with their Schur
square, and we provide conditions for an ideal to be such that its Schur square
has the projective cover of the trivial module as a direct summand. This has
particularly interesting consequences for group algebras of p-groups over
fields of characteristic p.
