The Assmus{Mattson theorem is a method to nd designs in linear codes over a nite eld. The purpose of this paper is to give an analogue of this theorem for Z 4 -codes by using the harmonic weight enumerator which is introduced by Bachoc. This theorem can nd some 5-designs in the lifted Golay code over Z 4 which were discovered previously by other methods. Index Terms { Z 4 -codes, Assmus{Mattson...

In this paper, we describe the properties of the i-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every q-ary code of length m and minimum distance 5 (for q = 3 the minimum distance is 3) can be embedded in a q-ary 1-perfect code of length n = (qm−1)/(q−1). It is also shown that every binary code of length m+k an...

We study odd and even Z2Z4 formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.

The paper proves that there exists an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov (2007) are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the ...

A binary code is called a superimposed cover-free (s, ℓ)-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of ℓ sets is covered by the union of s others. A binary code is called a superimposed list-decoding s L-code if the code is identified by the incidence matrix of a family of finite sets in which the union of any s sets can cover not ...