Ordered Binary Decision Diagrams and Minimal Trellises John La erty

Ordered binary decision diagrams (OBDDs) are graph-based data structures for representing Boolean functions. They have found widespread use in computer-aided design and in formal veri cation of digital circuits. Minimal trellises are graphical representations of error-correcting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphical models, as follows. Let C be a binary code of length n, and let fC (x1; : : : ; xn) be the Boolean function that takes the value 0 at x1; : : : ; xn if and only if (x1; : : : ; xn)2 C . Given this natural one-to-one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code C with minimum distance d > 1 is isomorphic to the singleterminal OBDD for its Boolean indicator function fC (x1; : : : ; xn). Prior to this result, the extensive research during the past decade on binary decision diagrams { in computer engineering { and on minimal trellises { in coding theory { has been carried out independently. As outlined in this work, the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines.