Olof Heden

A survey of perfect codes

The first examples of perfect

Maximal partial spreads and the modular n -queen problem II

e

On the Ranks and Kernels Problem for Perfect Codes

-error correcting

On the Structure of Symmetry Groups of Vasil'ev Codes

q

A Maximal Partial Spread of Size 45 in PG(3,7)

-ary codes were given in the 1940s by Hamming and Golay. In 1973 Tietavainen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect

A binary perfect code of length 15 and codimension 0

q

A Full Rank Perfect Code of Length 31

-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem. This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

On the reconstruction of perfect codes

We prove that if q + 1 E 8 or 16 (mod 24) then, for any integer n in the interval (q2 + 1)/2 + 3 < n < (Sq’ + 4q + 7)/8, there is a maximal partial spread of size n in PG(3, q).

On the structure of non-full-rank perfect

A construction is proposed which, for n large enough, allows one to build perfect binary codes of length n and rank r, with kernel of dimension k, for any admissible pair (r, k) within the limits of known bounds.

q

The structure of symmetry groups of Vasil’ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil’ev code of length n is always nontrivial; for codes of rank n − log(n + 1) +1, an attainable upper bound on the order of the symmetry group is obtained.