Altera Reed-Solomon Compiler - Manual (Page 20)

3–2 Chapter 3: Functional Description
Background
Reed-Solomon Compiler December 2014 Altera Corporation
User Guide
For example, for the following information:
a is a root of the binary primitive polynomial x 8 + x 7 + x 2 + x + 1
i0 = 120
You can calculate the following parameters:
■ R – 1 = 3
■ a = 1 (

is to the power 1 times i)
The field polynomial can be obtained by replacing x with 2, thus:
2
8 + 2 7 + 2 2 +2 + 1 = 391
Erasures
In normal operation, the RS decoder detects and corrects symbol errors.
The number of symbol errors that can be corrected, C, depends on the number of
check symbols, R and is given by C  R/2.
If the location of the symbol errors is marked as an erasure, the RS decoder can correct
twice as many errors, so C  R.
1
Erasures are symbol errors with a known location.
External circuitry identifies which symbols have errors and passes this information to
the decoder using the eras_sym signal. The eras_sym input indicates an erasure (when
the erasures-supporting decoder option is selected).
The RS decoder can work with a mixture of erasures and errors.
A codeword is correctly decoded if (2e + E)  R
where:
e = errors with unknown locations
E = erasures
R = number of check symbols.
For example, with ten check symbols the decoder can correct ten erasures, or five
symbol errors, or four erasures and three symbol errors.
1
If the number of erasures marked approaches the number of check symbols, the
ability to detect errors without correction (decfail asserted) diminishes. Refer to
Table 3–1 on page 3–4 .
Shortened Codewords
A shortened codeword contains fewer symbols than the maximum value of N, which
is 2
m –1. A shortened codeword is mathematically equivalent to a maximum-length
code with the extra data symbols at the start of the codeword set to 0.
For example, (204,188) is a shortened codeword of (255,239). Both of these codewords
use the same number of check symbols, 16.
3
g(x) = (x –  i + i
0
)
i = 0