## Repetition Code with Arbitrary Length

A linear [*s*, 1, *s*]-code C over **F**_{b} exists for all *s* ≥ 1. Its dual A = C^{⊥} is a linear orthogonal array OA(*b*^{s−1}, *s*,**F**_{b}, *s*−1).

### The Orthogonal Array (Hyperplane)

The orthogonal array A can be identified with a hyperplane in general position in **F**_{b}^{s}. 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 **F**_{b}.

### The Linear Code (Repetition Code)

The code C is constructed as

*x*,…,

*x*) :

*x*∈

**F**

_{b}} ⊂

**F**

_{b}

^{s}.

Therefore the 1×*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*,**F**_{b}1), 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 ℤ_{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.

### See Also

### References

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

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Repetition Code with Arbitrary Length.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CRepetition-inf.html