Block codes - Codes and Turbo Codes - page 146

where Δ 2 is the determinant of the system with two equations.

Δ 2 = S 2 + S 1 S 3

Finally, for an RS code correcting three errors ( t =3 ), the relation (4.36) with
t =3 and q =1 , 2 , 3 leads to the following system of three equations:

σ 1 S 3 + σ 2 S 2 + σ 3 S 1 = S 4
σ 1 S 4 + σ 2 S 3 + σ 3 S 2 = S 5
σ 1 S 5 + σ 2 S 4 + σ 3 S 3 = S 6

The resolution of this system enables us to determine coecients σ 1 , σ 2 and σ 3
of the error locator polynomial.
σ 1 = Δ 3 S 1 S 3 S 6 + S 1 S 4 S 5 + S 2 S 6 + S 2 S 3 S 5 + S 2 S 4 + S 3 S 4

σ 2 = Δ 3 S 1 S 4 S 6 + S 1 S 5 + S 2 S 3 S 6 + S 2 S 4 S 5 + S 3 S 5 + S 3 S 4
σ 3 =

(4.39)

S 2 S 4 S 6 + S 2 S 5 + S 3 S 6 + S 4

1
Δ 3

where Δ 3 is the determinant of the system with three equations.

Δ 3 = S 1 S 3 S 5 + S 1 S 4 + S 2 S 5 + S 3

Implementation of Peterson's decoder for an RS code with parameter t =3

1. Calculate the 2 t syndromes S j : S j = r ( α j )

2. Determine the number of errors:

• Case (a) S j =0 , ∀ j : no detectable error.

• Case (b) Δ 3 =0 : presence of three errors.

• Case (c) Δ 3 =0 and Δ 2 =0 : presence of two errors.

• Case (d) Δ 3 =Δ 2 =0 and S 1 =0 : presence of one error.

3. Calculate the error locator polynomial σ d ( x )

• Case (b) Use (4.39)

• Case (c) Use (4.38)

•

Case (d) Use (4.37)

4. Look for the roots of σ d ( x ) in field F q

5. Calculate the error coecients e i