29.8. 29.8 Superperfect arrays

This section deals with the construction of infinite De Bruijn arrays (superperfect arrays).

Definition 29.5 Let , be positive integers (), . A -array (or De Bruijn array) is a periodic array with elements from , in which all of the different -ary arrays appear exactly ones.

Definition 29.6 Let , and be positive integers, . A -array (or infinite De Bruijn array) is a -ary infinite array with elements from whose beginning parts of length as periodic arrays are -arrays for .

The following theorems [ 125 ] give construction results.

Theorem 29.7 (Iványi [ 125 ]) For any , and there are and such that -array exists.

Proof. See [ 125 ].

Theorem 29.8 (Iványi [ 125 ]) For any odd and and also for any even and there is an such that a -array exists.

Proof. See [ 125 ].

Theorem 29.8 does not cover the case when is even and . No - and -arrays exist [ 124 ].