ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 4, pp. 319–329.
Pleiades Publishing, Inc., 2006.
Original Russian Text
I. Landjev, A. Rousseva, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 4, pp. 65–76.
An Extension Theorem for Arcs and Linear Codes
and A. Rousseva
Institute of Mathematics and Informatics, BAS, Soﬁa, Bulgaria
Faculty of Mathematics and Informatics, Soﬁa University, Bulgaria
Received November 15, 2005; in ﬁnal form, May 27, 2006
Abstract—We prove the following generalization to the extension theorem of Hill and Lizak:
For every nonextendable linear [n, k, d]
code, q = p
i≡0,d (mod q)
where q + r(q) + 1 is the smallest size of a nontrivial blocking set in PG(2,q). This result is
applied further to rule out the existence of some linear codes over F
meeting the Griesmer
be the vector space of all n-tuples over the q-element ﬁeld F
.Aq-ary linear code of
length n and dimension k is a k-dimensional subspace C of F
. The minimum distance of C is
deﬁned as the minimum Hamming distance between any two diﬀerent words of C. A linear code
of length n,dimensionk, and minimum distance d is referred to as an [n, k, d]
code. Given an
[n, k, d]
, w =0, 1,...,n, the number of codewords of Hamming weight w.
The sequence A
,... is called the spectrum of C. We refer to classical books on
coding theory (like [1–4]) for further notions and results about linear codes over ﬁnite ﬁelds. Basic
results about special pointsets in projective geometries over ﬁnite ﬁelds can be found in [5–7].
Let C be an [n, k, d]
code. The code C is said to be extendable if there exists an [n +1,k,d+1]
code D such that C is equivalent to some punctured code obtained from D. A well-known fact is
that the existence of an [n, k, d]
code with odd d implies the existence of an [n +1,k,d+1]
by adding a parity-check coordinate; i.e., an [n, k, d]
code with odd d is always extendable. This
observation was generalized in [8, 9], where the following result was proved.
Theorem 1 . Let C be an [n, k, d]
code with gcd(d, q)=1and with A
=0for all i ≡ 0,
i ≡ d (mod q).ThenC is extendable to an [n +1,k,d+1]
This result was reﬁned in [10, 11].
In what follows, we prove a generalization of Hill’s theorem and employ this result to derive the
nonexistence of some linear codes with optimal parameters (in the sense of the Griesmer bound).
In this, we use the so-called geometric approach to linear codes. It has long been known that
linear codes of full length, i.e., codes with the property that no coordinate is identically zero in all
codewords, are equivalent to arcs in the appropriate projective geometries (cf. ). This makes it
possible to apply results about sets of points in projective geometries to linear codes and vice versa.
Supported in part by the NFNI, contract no. M-1405/2005.