ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 3, pp. 221–231.
⃝ Pleiades Publishing, Inc., 2009.
Original Russian Text
⃝ V.A. Zinoviev, T. Ericson, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 3, pp. 33–44.
Fourier-Invariant Pairs of Partitions of Finite
Abelian Groups and Association Schemes
V. A. Zinoviev
and T. Ericson
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received October 7, 2008; in ﬁnal form, May 14, 2009
Abstract—We consider partitions of ﬁnite abelian groups. We introduce the concept of
Fourier-invariant pairs and demonstrate that this concept is equivalent to the concept of an
association scheme in an abelian group. It follows that Fourier-invariant pairs of partitions
might be viewed as a very natural approach to abelian association schemes.
The well-known MacWilliams identities were originally formulated for linear codes and the
Hamming weight . In essence, the result is a linear transform connecting the weight distributions
of a code and its dual in terms of the so-called Krawtchouk polynomials. The result is extremely
useful both as a tool for studying linear codes and as a bound for unrestricted codes (linear or not)
by the so-called linear programming bound, introduced by Delsarte .
The result can be generalized in several ways. Linearity can be replaced by a more general con-
cept of ﬁnite abelian groups, and instead of Hamming weight it is possible to consider the partition
of an abelian groupinto subsets such that all elements of one subset have the same “weight.” This
leads in a natural way to so-called Fourier-invariant partitions of an abelian group [3, 4].
In the present paper, the concept of a Fourier-invariant partition is generalized to what we call
Fourier-invariant pairs of partitions. Examples are given, and some basic properties are developed.
In particular, we show that there is a close connection between Fourier-invariant pairs and abelian
association schemes . In fact, a Fourier-invariant pair of partitions can be used in order to
formulate an alternative deﬁnition of an abelian association scheme. We give two inﬁnite families
of Fourier-invariant pairs of partitions for arbitrary ﬁnite abelian groups. In opinion of the authors
of the present paper, the introduced Fourier-invariant pairs of partitions can simplify the search,
description, and construction of association schemes in ﬁnite abelian groups of arbitrary order.
2. PARTITIONS OF ABELIAN GROUPS
Let be a ﬁnite abelian group, written additively. We consider only ﬁnite groups. A collection
∣ ... ∣
of is called a partition if it satisﬁes the following properties:
(a) ∕= ⇒
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.