ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 1, pp. 21–30.
Pleiades Publishing, Inc., 2012.
Original Russian Text
I.E. Bocharova, F. Hug, R. Johannesson, B.D. Kudryashov, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48,
No. 1, pp. 26–36.
Dual Convolutional Codes
and the MacWilliams Identities
I. E. Bocharova
, R. Johannesson
, and B. D. Kudryashov
St. Petersburg State University of Information Technologies,
Mechanics and Optics (ITMO)
Lund University, Sweden
Received April 13, 2011; in ﬁnal form, December 8, 2011
Abstract—A recursion for sequences of spectra of truncated as well as tailbitten convolutional
codes and their duals is derived. The order of this recursion is shown to be less than or equal to
the rank of the weight adjacency matrix (WAM) for the minimal encoder of the convolutional
code. It is suﬃcient to know ﬁnitely many spectra of these terminated convolutional codes in
order to obtain an inﬁnitely long sequence of spectra of their duals.
Following , we deﬁne a rate R = b/c binary convolutional code over the ﬁeld F
as the image
set of the linear mapping represented by
where the code sequence v(D) and information sequence u(D)arec-andb-tuples over the ﬁeld of
Laurent series, v(D) ∈ F
((D)) and u(D) ∈ F
((D)), respectively, and the generator matrix G(D)
over the ﬁeld of rational functions F
Convolutional codes are often thought of as nonblock linear codes over a ﬁnite ﬁeld. Sometimes,
however, it is an advantage to regard a convolutional code as a block code over a certain inﬁnite
ﬁeld; in our case, as the F
(D)rowspaceofG(D) or, in other words, as a rate R = b/c block code
over the inﬁnite ﬁeld of Laurent series encoded by G(D). From this point of view it seems rather
natural that convolutional codes would have similar properties as block codes and satisfy proper
reformulations of theorems valid for block codes.
Let the free distance be denoted by d
. Then the path weight enumerator of a convolutional
encoder introduced by Viterbi  is the generating function T (W )=
Hamming weights of the paths which diverge from the all-zero path at the root in the trellis
representation of the encoder and terminate in the zero state, but do not merge with the all-zero
path until their termini. In the sequel we call the sequence n
, i =0, 1, 2,...,thefree distance
spectrum or Viterbi spectrum in order to distinguish it from the spectrum of block codes. It is
well known, starting with , that the MacWilliams identity does not hold for the free distance
spectra of convolutional encoders. In [4–6], MacWilliams-type identities were established, not for
the free distance spectrum but for the so-called weight adjacency matrix (WAM) . In particular,
Supported in part by the Swedish Research Council, Grant no. 621-2007-6281.