Repetition Code with Arbitrary Length
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
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,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 ℤ2 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: 2024-09-05.
http://mint.sbg.ac.at/desc_CRepetition-inf.html