Problems of Information Transmission, Vol. 40, No. 2, 2004, pp. 159–164. Translated from Problemy Peredachi Informatsii, No. 2, 2004, pp. 63–69.
Original Russian Text Copyright
2004 by Tokareva.
On Components of Preparata Codes
N. N. Tokareva
Novosibirsk State University
Received November 3, 2003; in ﬁnal form, February 19, 2004
Abstract—The paper considers the interrelation between i-components of an arbitrary Pre-
parata-like code P and i-components of a perfect code C containing P .Itisshownthateach
i-component of P can uniquely be completed to an i-component of C by adding a certain
number of special codewords of C. It is shown that the set of vertices of P in a characteristic
graph of an arbitrary i-component of C forms a perfect code with distance 3.
It is well known (see ) that there are only two nontrivial inﬁnite families of codes that are
both maximal and uniformly packed—these are the (closely related) families of perfect codes and
The paper reveals one more property reﬂecting the relation between Preparata codes and perfect
code containing them. The main subject of our study are components of Preparata codes and
perfect codes; the main tool is the component switching method.
By a component of a code we mean a subset of codewords allowing special-type transformations,
which change a code but preserve its parameters (code length, cardinality, and distance). Presently,
the switching method (method of independent transformations of diﬀerent components of a code)
plays one of the main roles in studying the properties of perfect codes (see [2–4]); however, it should
be noted that the method has never been applied to Preparata codes.
Each Preparata code is contained in a certain perfect code, which is unique (see ). The
converse is not true in general. If, by switching of an arbitrary i-component (for the deﬁnition, see
Section 2), we pass from a Preparata code P to a Preparata code P
, what is the relation between
perfect codes C and C
that contain P and P
We show that any i-component of a Preparata code can uniquely be completed to an i-component
of a perfect code, and a switch of an i-component in the Preparata code induces in the ambient
perfect code a switch of the completed i-component only (Theorem 1). Thus, codes C and C
obtained from each other by switching involving one coordinate i only. As a consequence, we obtain
estimates for the cardinality of a minimal i-component of a Preparata code. We also show that
the set of vertices of a Preparata code in a characteristic graph of an arbitrary i-component of the
ambient perfect code forms a perfect code with distance 3 (Theorem 2).
2. NECESSARY DEFINITIONS AND STATEMENTS
Consider the n-dimensional vector space E
over Galois ﬁeld GF (2) equipped with the Hamming
metric: the distance d(x, y) between vectors x and y (we will also call them vertices) is the number
of positions in which they diﬀer. The weight w(x)ofavertexx is d(x, 0
), where 0
zero vertex, i.e., the vertex with all coordinates equal to zero. A set C ⊆ E
is called a code of
2004 MAIK “Nauka/Interperiodica”