TY - JOUR

T1 - Tail-biting trellises of block codes

T2 - Trellis complexity and viterbi decision complexity

AU - Reuven, Ilan

AU - Be'ery, Yair

PY - 1999

Y1 - 1999

N2 - Tail-biting trellises of linear and nonlinear block codes are addressed. We refine the information-theoretic approach of a previous work on conventional trellis representation, and show that the same ideas carry over to tail-biting trellises. We present lower bounds on the state and branch complexity profiles of these representations. These bounds are expressed in terms of mutual information between different portions of the code, and they introduce the notions of superstates and superbranches. For linear block codes, our bounds imply that the total number of superstates, and respectively superbranches, of a tail-biting trellis of the code cannot be smaller than the total number of states, and respectively branches, of the corresponding minimal conventional trellis, though the total number of states and branches of a tail-biting trellis is usually smaller than that of the conventional trellis. We also develop some improved lower bounds on the state complexity of a tail-biting trellis for two classes of codes: the first-order Reed-Muller codes and cyclic codes. We show that the superstates and superbranches determine the Viterbi Decision complexity of a tail-biting trellis. Thus, the computational complexity of the maximum-likelihood Decision of linear block codes on a tail-biting trellis, using the Viterbi algorithm, is not smaller than that of the conventional trellis of the code. However, tail-biting trellises are beneficial for suboptimal and iterative Decision techniques.

AB - Tail-biting trellises of linear and nonlinear block codes are addressed. We refine the information-theoretic approach of a previous work on conventional trellis representation, and show that the same ideas carry over to tail-biting trellises. We present lower bounds on the state and branch complexity profiles of these representations. These bounds are expressed in terms of mutual information between different portions of the code, and they introduce the notions of superstates and superbranches. For linear block codes, our bounds imply that the total number of superstates, and respectively superbranches, of a tail-biting trellis of the code cannot be smaller than the total number of states, and respectively branches, of the corresponding minimal conventional trellis, though the total number of states and branches of a tail-biting trellis is usually smaller than that of the conventional trellis. We also develop some improved lower bounds on the state complexity of a tail-biting trellis for two classes of codes: the first-order Reed-Muller codes and cyclic codes. We show that the superstates and superbranches determine the Viterbi Decision complexity of a tail-biting trellis. Thus, the computational complexity of the maximum-likelihood Decision of linear block codes on a tail-biting trellis, using the Viterbi algorithm, is not smaller than that of the conventional trellis of the code. However, tail-biting trellises are beneficial for suboptimal and iterative Decision techniques.

KW - Maximum-likelihood decision

KW - Mutual information

KW - Tail-biting trellis

KW - Trellises

KW - Viterbi algorithm

UR - http://www.scopus.com/inward/record.url?scp=1842448238&partnerID=8YFLogxK

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AN - SCOPUS:1842448238

SN - 0916-8508

VL - E82-A

SP - 2043

EP - 2051

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

IS - 10

ER -