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Error Correcting Codes: Combinatorics, Algorithms and Applications
(Fall 2007)
Lecture 15: GilbertVarshamov Bound
October 2, 2007
Lecturer: Atri Rudra
Scribe: ThanhNhan Nguyen
In the previous lectures, we have only seen upper bounds on the rate of a code (given a ﬁxed
relative distance). In this lecture, we study our ﬁrst (and only) lower bound on rate of a code.
1
GilbertVarshamov Bound
In today’s lecture we will prove the following result
Theorem 1.1.
Let
q
≥
2
. For every
0
≤
δ <
1

1
q
, and
0
< ε
≤
1

H
q
(
δ
)
, there exists a code
with rate
R
≥
1

H
q
(
δ
)

ε
, and relative distance
δ
.
The above result was proved for general codes by Gilbert (Section 1.1) and for linear codes by
Varshamov (Section 1.2). Hence, the bound is called the GilbertVarshamov bound.
1.1
Gilbert Construction
Gilbert proved Theorem 1.1 by the following greedy construction (where
d
=
δn
)
(i) Start with the empty code:
C
← ∅
(ii) If there exists a
v
∈
[
q
]
n
such that
Δ(
v
,
c
)
≥
d
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