We list the most commonly used ones below [6].
Hamming weight, w(c). This measurement is the number of nonzero bit locations in the code word
vectors, c.
Hamming distance, d(c1,c2). This measurement is the number of elements (or bits) in which the
two code words, c1 and c2, differ.
Minimum distance, dmin. This measurement is the smallest value of the Hamming distance, d(ci,cj)
for i not equal to j.
Error correcting capability, t. This measurement is the number of bit errors that the FEC code
word can correct.
(5.3) t jdmin 1
2 k
Error detecting capability, e. This measurement is the number of bit errors the FEC code word can
detect.
(5.4)
Let us present the mathematical foundation necessary to understand the FEC encoding operations.
Consider the information bits at the encoder input to be denoted as.
(5.5)
And the output of the FEC encoder is a vector defined below.
(5.6)
The generation of the code word is represented as
(5.7)
where G is commonly called the FEC generation matrix of the code. This provides a one-to-one mapping
of the input and output words. The elements of this matrix are given below.
(5.8)
For what is typically called systematic codes, the generator matrix can be modified and will have the
following form.
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