## Extended Reed–Solomon Code

For every field Fb and k with 0 ≤ kb there is a linear orthogonal array with bk runs, b + 1 factors, strength k and minimum distance b + 2−k. In other words, it can be interpreted either as a linear OA(bk, b + 1, b, k) or as a linear [b + 1, k, b + 2−k]-code.

### Construction of the Reed-Solomon Code

Let x1,…, xb denote the b elements of Fb. The Reed-Solomon code RS(n, b) for n = 0,…, b is given by

RS(n, b) := {(f (xi))i=1,…, b  :  fFb[X] with deg f < n}.

It is a linear [b, n, bn + 1]-code as well as a linear OA(bn, b,Fb, n).

Reed-Solomon codes over a given field Fb are all subcodes of each other, i.e., we have RS(n, b) ⊂ RS(n + 1, b). The dual code of a Reed-Solomon code is again a Reed-Solomon code, namely RS(n, b) = RS(bn, b).

Reed-Solomon codes can also be interpreted as algebraic-geometric codes defined by the rational function field Fb(x).

The Reed-Solomon code as an orthogonal array was first considered in , where the result on the minimum strength was established. That the same construction leads to a code with minimum distance sn + 1 is due to .

### The Extended Reed-Solomon Code

The extended Reed-Solomon code RSe(n, b) for n = 0,…, b is given by

RSe(n, b) := {(f (x1),…, f (xb), f (∞))  :  fFb[X]withdeg f < n},

where f (∞) denotes the coefficient of Xn−1 in f.

RSe(n, b) can also be obtained by extending RS(n – 1, b) ⊂ RS(n, b) with construction X and the [1, 1, 1]-code without redundancy. RSe(n, b) is a linear [b + 1, n, bn + 2]-code as well as a linear OA(bn, b + 1, b, n).

Results for extended Reed-Solomon codes were established by , , and .

### Optimality

Reed-Solomon codes and extended Reed-Solomon codes meet the Singleton bound with equality, and are therefore MDS-codes. Viewed as an orthogonal array, they are OAs with index unity.

### Special Cases

• Generalization for arbitrary OOAs

• Reed-Solomon codes can be generalized to algebraic-geometric codes by allowing a more general class of functions instead of polynomials.

• [6, Chapter 4]

### References

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