Proof of a conjecture of McEliece regarding the expansion index of the minimal trellis

We prove a conjecture of McEliece, establishing that for each fixed order of positions of a linear code C, the minimal trellis minimizes the quantity |E|-|V|+1, where |E| and |V| stand for the number of edges and vertices in the trellis, respectively. As a consequence, it follows that the minimal trellis uniquely minimizes the total number of operations required for Viterbi decoding of a given code. Moreover, we show that these results extend to the general class of rectangular codes (namely, the class of codes that admit a biproper trellis presentation), which includes all group codes and many other useful nonlinear codes.

[1]  Narsingh Deo,et al.  Graph Theory with Applications to Engineering and Computer Science , 1975, Networks.

[2]  Alexander Vardy,et al.  Asymptotically good codes have infinite trellis complexity , 1995, IEEE Trans. Inf. Theory.

[3]  G. David Forney,et al.  Coset codes-II: Binary lattices and related codes , 1988, IEEE Trans. Inf. Theory.

[4]  R.J. McEliece.  The Viterbi decoding complexity of linear block codes , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[5]  Alexander Vardy,et al.  Lower bounds on trellis complexity of block codes , 1995, IEEE Trans. Inf. Theory.

[6]  Shu Lin,et al.  On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes , 1993, IEEE Trans. Inf. Theory.

[7]  Jan C. Willems,et al.  Models for Dynamics , 1989 .

[8]  Douglas J. Muder.  Minimal trellises for block codes , 1988, IEEE Trans. Inf. Theory.

[9]  Mitchell D. Trott,et al.  The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders , 1993, IEEE Trans. Inf. Theory.

[10]  Robert J. McEliece,et al.  On the BCJR trellis for linear block codes , 1996, IEEE Trans. Inf. Theory.

[11]  Frank R. Kschischang.  The trellis structure of maximal fixed-cost codes , 1996, IEEE Trans. Inf. Theory.

[12]  Alexander Vardy,et al.  Optimal sectionalization of a trellis , 1996, IEEE Trans. Inf. Theory.

[13]  John Cocke,et al.  Optimal decoding of linear codes for minimizing symbol error rate (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[14]  Samuel Dolinar,et al.  Trellis decoding complexity of linear block codes , 1996, IEEE Trans. Inf. Theory.

[15]  G. David Forney,et al.  Dimension/length profiles and trellis complexity of linear block codes , 1994, IEEE Trans. Inf. Theory.

[16]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[17]  Andrew J. Viterbi,et al.  Error bounds for convolutional codes and an asymptotically optimum decoding algorithm , 1967, IEEE Trans. Inf. Theory.

[18]  Alexander Vardy,et al.  Maximum-Likelihood Soft Decision Decoding of Bch Codes , 1993, Proceedings. IEEE International Symposium on Information Theory.

[19]  Frank R. Kschischang,et al.  On the trellis structure of block codes , 1994, IEEE Trans. Inf. Theory.

[20]  A. D. Kot,et al.  On the Construction and Dimensionality of Linear Block Code Trellises , 1993, Proceedings. IEEE International Symposium on Information Theory.

[21]  Jack K. Wolf,et al.  Efficient maximum likelihood decoding of linear block codes using a trellis , 1978, IEEE Trans. Inf. Theory.