Foreword
This chapter is based on lecture notes from the course CSE 545– Error-Correcting Codes: Combinatorics, Algorithms and Applications taught by Atri Rudra at University at Buffalo, SUNY.
This version is dated February 29, 2012. For the latest version, please go to
http://www.cse.buffalo.edu/ atri/courses/coding-theory/book/
The material in this chapter is supported in part by the National Science Foundation under
CAREER grant CCF-0844796. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the
National Science Foundation (NSF).
©Atri Rudra, 2012.
This work is licensed under the Creative Commons Attribution-NonCommercialNoDerivs 3.0 Unported License. To view a copy of this license, visit
http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 444
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Appendix A
Notation Table
R
¬E
log x
Σm
v
0
ei
vS
⟨u, v⟩
[a, b]
[x]
n
Σ
q
k
R
∆(u, v)
d
w t (v)
B (x, r )
(n, k, d )Σ
(n, k, d )q
[n, k, d ]q
Fq
F∗
Fm×N
q
R(C )
δ(C )
U
The set of real numbers
Negation of the event E
Logarithm to the base 2
Vectors of length m with symbols from Σ
A vector
The all zero vector
The i th standard vector, i.e. 1 in position i and 0 everywhere else
Vector v projected down to indices in S
Inner-product of vectors u and v
{x ∈ R|a ≤ x ≤ b}
The set {1, . . . , x}
Block length of a code
Alphabet of a code
q = |Σ|
Dimension of a code
Rate of a code
Hamming distance between u and v
Minimum distance of a code
Hamming weight of v
Hamming ball of radius r centered on x
A code with block length n, dimension k, distance d and alphabet Σ
A code with block length n, dimension k, distance d and alphabet size q
A linear (n, k, d )q code
The finite field with q elements (q is a prime power)
The set of non-zero elements in the field F
The set of all m × N matrices where each entry is from Fq
Rate of a code family C
Relative distance of a code family C
The uniform distribution
183
Section 1.2
Definition 1.2.1
Definition 1.2.1
Definition 1.2.1
Definition 1.2.2
Definition 1.2.3
Definition 1.4.1
Definition 1.4.2
Definition 1.5.1
Definition 1.6.1
Definition 1.7.1
Definition 1.7.1
Definition 2.0.4
Section 2.1
Definition 2.5.1
Definition 2.5.1
Definition 3.1.1
E[V ]
1E
H q (x)
H q−1 (y)
deg(P )
Fq [X ]
Expectation of a random variable V
Indicator variable for event E
x logq (q − 1) − x logq x − (1 − x) logq (1 − x)
Unique x ∈ [0, 1 − 1/q] such that H q (x) = y
Degree of polynomial P (X )
The set of all univariate polynomials in X over Fq
184
Definition 3.1.2
Section 3.1
Definition 3.3.1
Section 3.3.2
Definition 5.1.2
Section 5.1