Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (GF(22m), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

A classification of quaternary Hermitian self-dual codes of length 20 is given. Using this classification, a classification of extremal quaternary Hermitian self-dual codes of length 22 is also given.

In this correspondence, we give some restrictions on weight enumerators of singly even self-dual [n,n/2,d] codes whose shadows have minimum weight d/2. As a consequence, we determine the weight enumerators for which there is an extremal singly even self-dual [40,20,8] code and an optimal singly even self-dual [50,25,10] code

Previously, Type II codes over F/sub 4/ have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence.

We study Type IV self-dual codes over the commutative rings of order 4. Gleason-type theorems of Type IV codes and their shadow codes are investigated. A mass formula of Type IV codes over these rings are given. We give a classification of Type TV codes over Z/sub 4/ and F2+uF/sub 2/ for reasonable lengths. We also construct a number of optimal Type TV codes.