1. Shuffle Exchange Network and de Bruijn’s Graph Shuffle Exchange graph 000 001 010011 100101 110111 00 01 10 11 Merge exchange into a single node De Bruijn Graph (label: shift left and add the label) 0 00 1 0 1 1 1

2. 00 01 10 11 1 0 1 0 0 11 0 Same Graph, Another labeling on edges node x 1 x 0  x 0 (x 1  label)

3.  f 001011 100110 000101010111 1 0 1 1 1 1 1 1 1 0 0 0 0 0 f is either 0 or 1 For 0: shift 1: complement Note that each complete cycle of shift register corresponds to a HC of de Bruijns Graph

4. 001 000 00. 

5.  0 0 1 Shift Register 001011 100110 000101010111 001 011 111 110 101 010 100 001 => DeBruijn sequence 001 010 101 011 111 110 100 001 0 01  011 100110 000101010111 x 3 + x + 1 is irreducible x 3 + x 2 + 1 is irreducible

6.  0 0 1 Shift Register 001011 100110 000101010111 001 011 110 100 => degenerated cycle x 3 + x 2 + x + 1 ? = (x 2 +1)(x+1) not irreducible 

7.  For 4 bit 0001 0011 0111 1111 1110 1101 1010 0101 1011 0110 1100 1001 0010 0100 1000

8. Cycle decomposition based n  001011 100110 011 011  101  010

9. 000 1 0 0 0 01 1 1 Conventional labeling 00 1 1 1 1 0 0 0 0

10. 001011 100110 000101010111