# An extension theorem for arcs and linear codes

An extension theorem for arcs and linear codes We prove the following generalization to the extension theorem of Hill and Lizak: For every nonextendable linear [n, k, d] q code, q = p s , (d,q) = 1, we have $$\sum\limits_{i\not \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),}$$ 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 $$\mathbb{F}_4$$ meeting the Griesmer bound. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# An extension theorem for arcs and linear codes

, Volume 42 (4) – Jan 24, 2006
11 pages

/lp/springer_journal/an-extension-theorem-for-arcs-and-linear-codes-lhtPeLrab6
Publisher
Nauka/Interperiodica
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946006040041
Publisher site
See Article on Publisher Site

### Abstract

We prove the following generalization to the extension theorem of Hill and Lizak: For every nonextendable linear [n, k, d] q code, q = p s , (d,q) = 1, we have $$\sum\limits_{i\not \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),}$$ 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 $$\mathbb{F}_4$$ meeting the Griesmer bound.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jan 24, 2006

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