y produce the same set of powers-of-a sequences for their output representations. Furthermore, when one product has J inputs and another has K inputs, J, , K, the repre- sentation of the K-fold product can potentially contain all powers-of-a sequences which occur in the J-fold product, and more. The word "poten- tially" is used because multiple terms with the same time coefficient c(i) (5.170) are produced, and there is a small likelihood that the coefficients of these terms will add to zero. The order D of an NLFFL function is defined as the largest number of factors in any product from its sum-of-products representation. The follow- ing theorem is a consequence of the above arguments. THEOREM 5.10 (KEY'S BOUND [23]). The linear span L of a sequence {bn} produced by NLFFL of order D operating on the contents of an M-stage m-sequence generator is bounded by (5.172) Non-Linear Feed-Forward Logic Table 5.11 Cyclic equivalence class representatives of non-zero coefficients in the powers-of-a representation in Example 5.11. coefficients base-2 base-10 000001 000011 000101 000111 001011 001101 010101 If the logic function includes the possibility of complementation by the use of an additional constant input term "1" in its sum-of-products representa- tion, then the lower limit on the sum in Key's bound (5.172) must be reduced to zero. Example 5.11. Key's bound on the linear span L for the output of third- order logic operating on a six-stage m-sequence generator is (5.173) The linear span achieved under these conditions in Example 5.10 is 38. Binary 6-tuple representations of the time coefficients (5.170) occuring in {bn}'s powers-of-aexpansion in the example include the ones shown in Table 5.11 and their cyclic shifts. This list includes cyclically equivalent represen- tatives of all 6-tuples having weight 3 or less, with the exception of 001001 which in decimal is 9. It can be verified that don is produced eight different ways in the product-of-sums to sum-of-products conversion (5.164) in the example, and that the 