On a Characteristic of Subsets of Abelian groups
by
Boris V. Novikov
University of Kharkov, Ukrainå

Below G denotes a finite elementary Abelian 2-group, T is its subset containing 1. The following notion can be useful for studying non-linear codes:

Definition 1. We say that T has the defect <= n (def T <= n) if |T\aT| <= n for every a in T. An element a in T is called (n)-element if |T\aT| = n.

Evidently, def T <= |G\T| for every T subset G.

Definition 2. T is standard if | < T > | = T+ def T, where < T > is the subgroup generated by T.

Proposition 1. If def T=1 then T is standard.

Definition 3. Let def T=n. T is degenerate if for all distinct (n)-elements a, b in T their product ab does not belong to T.

Proposition 2. If def T=2 then T is either standard or degenerate.

This assertion is wrong if def T=3:

Example. Let |G|=16, G= < a, b, c, d > , T={1, a, b, c, d, ab, cd}. Then def T=3 and T is neither standard nor degenerate.

Date received: July 25, 2001