Problems of Information Transmission, Vol. 41, No. 2, 2005, pp. 91–104. Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 26–41.
Original Russian Text Copyright
2005 by Nogin.
Weight Functions and Generalized Hamming Weights
of Linear Codes
D. Yu. Nogin
Institute for Information Transmission Problems, RAS, Moscow
Received December 2, 2003; in ﬁnal form, November 16, 2004
Abstract—We prove that the weight function wt: F
→ Z on a set of messages uniquely
determines a linear code of dimension k up to equivalence. We propose a natural way to extend
the rth generalized Hamming weight, that is, a function on r-subspaces of a code C,toa
function on F
C. Using this, we show that, for each linear code C and any integer
r ≤ k =dimC, a linear code exists whose weight distribution corresponds to a part of the
generalized weight spectrum of C,fromtherth weights to the kth. In particular, the minimum
distance of this code is proportional to the rth generalized weight of C.
We start with a simple question; the answer to this question, though known in the binary
case [1, Section 3.6], is well forgotten. Let a linear [n, k]
code C be given; i.e., q
transmitted with the help of length-n words. Since the code is linear, it is natural to consider that
the set of messages also has a linear structure, i.e., is identiﬁed with the space F
. Assume now that
for each message we only know the weight of the codeword by which this message is transmitted
but not the codeword itself. If we know these weights only, is it possible to reconstruct the code
uniquely up to equivalence (i.e., permutation of coordinates and multiplying each coordinate by a
constant)? A simple observations shows that the answer is positive (cf. Section 2).
Thus, given all the weights, i.e., a nonnegative function F
→ Z (more precisely, a function
→ Z since the weight of the zero word is always zero, and proportional words have the
; here, P
PC, the isomorphism PF
PC is induced by an isomorphism
of linear spaces F
C deﬁned by choosing any generator matrix of the code), one can reconstruct
a code according to a certain rule (see formula (4) below) up to equivalence, or (which is the
same) reconstruct the projective system that corresponds to the code. In turn, a projective system
(a multiset in PC
) is a nonnegative function ν : PC
→ Z,whereν(h) is the multiplicity of a point
h ∈ PC
. In other words, there exists a transform which takes a function wt to a function ν.
If we apply this transform to an arbitrary function
→ Z which is not a weight function
of any code, then the function
→ Q obtained according to (4) will not necessarily be
integer and nonnegative. However, with an appropriate choice of constants a and b, the function
ν + b becomes integer and nonnegative, i.e., deﬁnes a projective system (code) whose weights are
easily computed from
wt, a,andb. This observation can be viewed as a way of constructing linear
codes. Of course, for an arbitrary initial function
wt, the multiplicities thus obtained can be very
Supported in part by the Russian Foundation for Basic Research, project no. 02-01-01041, and INTAS,
grant no. 00-738.
we denote an arbitrary projective space of (projective) dimension r;byPV
we denote the projectivization of a linear space V .
2005 Pleiades Publishing, Inc.