## Singleton Bound

Every orthogonal array OA(M, s, Sb, k) satisfies Mbk, which follows directly from the definition of orthogonal arrays. Therefore the index λ = M/bk = bt must be ≥ 1 and t must be non-negative.

The dual result for codes is less obvious and is known as Singleton bound . It states that every (s, N, d)-code over Fb satisfies Nbs−d+1. For linear codes, this implies that every linear [s, n, d]-code over Fb satisfies n + ds + 1.

### Codes and Orthogonal Arrays Meeting the Singleton Bound

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

The most important class of MDS-codes are extended Reed-Solomon codes and the codes obtained from the hyperoval as well as codes obtained from them by truncation. However, codes without redundancy (minimum distance d = 1), parity check codes (minimum distance d = 2), repetition codes (minimum distance d = s), and trivial codes (minimum distance d = s + 1) are also MDS-codes.

The MDS-code conjecture states that no other MDS-codes exist.

• [2, Theorem 1.11] and [3, Theorem 4.1]

• Generalization for arbitrary OOAs

• Corresponding result for nets and sequences

### References

  Richard C. Singleton.Maximum distance q-nary codes.IEEE Transactions on Information Theory, 10(2):116–118, April 1964.  F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977.  Jürgen Bierbrauer.Introduction to Coding Theory.Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004.MR2079734 (2005f:94001)