## (Dual) Plotkin Bound

An orthogonal array OA(M, s, Sb, k) can exist only if

Mbs1 – .

This inequality is trivially satisfied and consequently the theorem gives no information if sb(k + 1)/(b−1). For b = 2 the bound was first established in [1], the general result is given in [2] and [3]. [2] gives an elementary proof whereas in [3] the dual Plotkin bound is derived from the linear programming bound.

The dual of this bound is the Plotkin bound [4], which states that for all (s, N, d)-codes over Fb with bd > (b−1)s we have

N.

It is now known that both bounds are implied by the linear programming bound for orthogonal arrays. There also exists a generalization for OOAs with arbitrary depth.

### Application to Nets and Sequences

In [5, Theorem 4.4.14] the dual Plotkin bound is used for establishing the corresponding result for (t, t + k, s)-nets.

Niederreiter applies this result to (t, t + k, s + 1)-nets with k = ⌊ss/b⌋ + 1 and derives the following bound for (t, s)-sequences: Let tb(s) be the minimum t such that a (t, s)-sequence in base b can exist. Then it is shown in [6, Theorem 8] that

tb(s) ≥  sO(log s)

and therefore

.

This is a better bound than the one obtained using the Rao bound for orthogonal arrays, and for b > 2 also better than the one resulting from the generalized Rao bound for OOAs. However, stronger bounds can be derived for all bases from the generalized Plotkin bound for OOAs.