Singleton Bound for OOAs

For every ordered orthogonal array OOA(M, s, S_b, T , k) we have M ≥ b^k, which follows directly from the definition of OOAs. Therefore the index λ = M/b^k = b^t must be ≥ 1 and t must be non-negative.

The dual result for NRT-codes is less obvious and is known as Singleton bound for NRT-codes [1, Theorem 1]. It states that every ((s, T ), N, d)-code over F_b satisfies N ≤ b^sT-d+1. For linear NRT-codes this implies that every linear [(s, T ), n, d]-code over F_b satisfies n + d ≤ sT + 1.

OOAs and NRT-Codes Meeting the Singleton Bound

NRT-Codes meeting the Singleton bound with equality are called maximum distance separable NRT-codes or MDS-NRT-codes. The dual of a linear MDS-NRT-code is a linear OOA with index unity.

The most important class of MDS-NRT-codes are Reed-Solomon NRT-codes. However, NRT-codes without redundancy (minimum distance d = 1), embedded parity check codes (minimum distance d = 2), repetition codes (minimum distance d = sT ), and trivial codes (minimum distance d = sT + 1) are also MDS-NRT-codes.

No other MDS-NRT-codes or OOAs with index unity are known.

See also

• For T = 1 the special case for orthogonal arrays and linear codes is obtained

• For T →∞ the special case for nets is obtained

• Corresponding result for sequences

References

[1]	Michael Yu. Rosenbloom and Michael A. Tsfasman. Codes for the m-metric. Problems of Information Transmission, 33:55–63, 1997.

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. “Singleton Bound for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_OBoundTrivialT.html