ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 1, pp. 10–29.
Pleiades Publishing, Inc., 2006.
Original Russian Text
V.A. Zinoviev, D.V. Zinoviev, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 1, pp. 13–33.
Vasil’ev Codes of Length n =2
and Doubling of
Steiner Systems S(n, 4, 3) of a Given Rank
V. A. Zinoviev, D. V. Zinoviev
Institute for Information Transmission Problems, Moscow
Received October 5, 2004; in ﬁnal form, November 8, 2005
Abstract—Extended binary perfect nonlinear Vasil’ev codes of length n =2
systems S(n, 4, 3) of rank n−m over F
are studied. The generalized concatenated construction
of Vasil’ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of
an arbitrary rank r over F
. We prove that any Steiner system S(n =2
, 4, 3) of rank n − m
can be obtained by this doubling construction and is formed by codewords of weight 4 of these
Vasil’ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of
all 12 nonequivalent Vasil’ev codes of length 16 are found. There are exactly 15 nonisomorphic
systems S(16, 4, 3) of rank 12 over F
, and they can be obtained from codewords of weight 4 of
the extended Vasil’ev codes. Orders of the automorphism groups of all these Steiner systems
One of interesting open problems of algebraic coding theory is the classiﬁcation of nonlinear
binary perfect codes with the Hamming-code parameters. An interesting class of such codes is the
class of Vasil’ev codes . According to Hergert , there are 19 nonequivalent Vasil’ev codes
of length 15 (including the linear code); according to Malyugin , there are 13 nonequivalent
extended Vasil’ev codes of length 16 (also including the linear code). Another interesting question
about these codes concerns their automorphism groups, which are not known even for the length
n = 16. In  there are some upper and lower bounds for the order of the automorphism groups
of Vasil’ev codes of any length n =2
in the case where automorphisms are only permutations of
In , Malyugin classiﬁed 370 nonlinear perfect codes of length 15 obtained from the Hamming
code by simultaneous shifts of components. In particular, orders of the automorphism groups
of these 18 nonequivalent nonlinear Vasil’ev codes of length 15 are found. Phelps  listed all
nonequivalent Solov’eva–Phelps codes [6,7] of length 16; their number was found to be 963. In ,
all extended perfect codes of length 16 and of ranks at most 13 were classiﬁed (their number is 285).
All perfect binary codes of length 15 and of ranks at most 13 were found in  (their number is 777).
Another open problem of combinatorial design theory is the classiﬁcation of nonisomorphic
Steiner systems S(16, 4, 3) [10–12]. In our previous papers [13,14], we enumerated all such systems of
. In particular, it was proved that there are exactly 15 such nonisomorphic
systems of rank 12, and 4131 nonisomorphic systems of rank 13.
The purpose of this paper is to ﬁnd full automorphism groups of all the 12 nonequivalent Vasil’ev
codes of length 16 and all the 15 corresponding nonisomorphic Steiner systems S(16, 4, 3), which can
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.