ON THE OPTIMUM BIT ORDERS WITH RESPECT TO THE STATE COMPLEXITY OF TRELLIS DIAGRAMS FOR BINARY LINEAR CODES

It was shown earlier that for a punctured Reed-Muller (PM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagram is much greater than that for an equivalent code obtained by a proper permutation on the bit positions. To find a permutation on the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is an interesting and challenging problem. This permutation problem is related to the generalized Hamming weight hierarchy of a code, and is shown that for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to Wei. The state complexity of trellis diagram for the extended and permuted (64,24) BCH code is discussed.