## Repetition Code

A linear [s, 1, s]-code C over Fb exists for all s ≥ 1. Its dual A = C is a linear orthogonal array OA(bs−1, s,Fb, s−1).

### The Orthogonal Array (Hyperplane)

The orthogonal array A can be identified with a hyperplane in general position in Fbs. Its generator matrix is a non-singular (s – 1)×s matrix.

If A is interpreted as a linear code, it is the [s, s−1, 2]-parity-check code over Fb.

### The Linear Code (Repetition Code)

The code C is constructed as

C = {(x,…, x)  :  xFb} ⊂ Fbs.

Therefore the s matrix (1,…, 1) is a generator matrix of C. C can also be constructed as an s-times juxtaposition or code repetition of the [1, 1, 1]-code without redundancy.

When interpreted as an orthogonal array, C is an OA(b, s,Fb1), the line-OA.

### Optimality

Repetition-codes meet the Singleton bound with equality and are therefore MDS-codes. Alternatively, their dual OAs are OAs with index unity.

Repetition codes over 2 are also perfect or nearly perfect codes, depending whether s is odd or even, because they meet the Hamming bound for s odd and can be truncated to perfect repetition codes with s odd if s is even. Alternatively, their dual OAs are (nearly) tight.

• [1, Chapter 1: Example Code #2 and Problem (4)]

• Generalization for arbitrary OOAs

### References

 [1] F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977.