Cryptography Reference
In-Depth Information
Chapter 4
Block codes
Block coding involves associating with a data block d of k symbols coming from
the information source, a block c , called the codeword, of n symbols with n
k .
The ( n
k ) is the amount of redundancy introduced by the code. Knowledge of
the coding rule at reception enables errors to be detected and corrected, under
certain conditions. The ratio k / n is called the coding rate of the code.
The message symbols of the information d and of the codeword c take their
values in a finite field F q with q elements, called a Galois field, whose main
properties are given in the appendix to this chapter. We shall see that for most
codes, the symbols are binary and take their value in the field F 2 with two
elements (0 and 1). This field is the smallest Galois field.
The elementary addition and multiplication operations in field F 2 are re-
sumed in Table 4.1.
a
b
a
+
b
ab
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
Table 4.1 - Addition and multiplication in the Galois field
F 2
A block code of length n is an application g of the set F q towards the set
F q that associates a codeword c with any block of data d .
F q
F q
g :
d
c = g ( d )
The set of q k
codewords generally constitutes a very reduced subset of F q .
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