Entropy/length profiles, bounds on the minimal covering of bipartite graphs, and trellis complexity of nonlinear codes

Abstract

In this paper, the trellis representation of nonlinear codes is studied from a new perspective. We introduce the new concept of entropy/length profile (ELP). This profile can be considered as an extension of the dimension/length profile (DLP) to nonlinear codes. This elaboration of the DLP, the entropy/length profiles, appears to be suitable to the analysis of nonlinear codes. Additionally and independently, we use wellknown information-theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs. We use these bounds in conjunction with the ELP notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of (nonlinear) block codes represented by any trellis diagram. We lay down no restrictions on the trellis structure, and we do not confine the scope of our results to proper or one-to-one trellises only. The basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords. We also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords. This approach is pursued further to derive similar bounds on the branch complexity profile. To the best of our knowledge, the proposed upper bounds are the only upper bounds that address nonlinear codes. The novel lower bounds are tighter than the existing bounds. The new quantities and bounds reduce to well-known results when applied to linear codes.

Nonlinear Product Codes and Their Low Complexity Iterative Decoding

“…In Table 4, |E| and |V| denote the total number of edges and the total number of states, respectively. Minimal trellises for nonlinear codes are generally computationally intractable, and the solutions leading to this minimization may result in improper or unobservable trellises [26]. State complexity comparison of NR code trellises is shown wf 2i+1 Firstly, a nonlinear product code composed by the (16, 8, 6) NR code as a row code and the (4, 3, 2) SPC code as a column code is designed for the sake of simplicity.…”

Ordered binary decision diagrams and minimal trellises

“…This function has a biproper OBDD, shown in Figure 5, even though C/ is not rectangular under any ordering. [33,47,53,54,63,65] has been devoted to lower bounds on the size of the minimal trellis for a given code, under all possible permutations of the time axis. Here, we translate some of these bounds into the language of binary decision diagrams.…”

“…Notably, this bound limits the size of the smallest OBDD that can be obtained under all possible orderings of the variables xi,...,x n . The bound was proved by Reuven and Be'ery[65] in the context of Let f(x l ,...,x n ) be a Boolean function such that d(C f ) > 1. Then the number of vertices at level i in the OBDD for f{x u ...,x n ) is lower bounded by…”

The weighted coordinates bound and trellis complexity of block codes and periodic packings

“…Measures of trellis complexity of block codes over a fixed alphabet set are bounded by the entropy/length profile (ELP) [18] which extends the dimension/length profile (DLP) of linear codes [7] to nonlinear codes. Several studies have addressed the problem of finding efficient permutations that meet the DLP bound, and hence minimize measures of trellis complexity (e.g., [3], [12], [13]).…”

“…There is no measure equivalent to the DLP and ELP for block codes whose symbols are taken from alphabets of different sizes, such as Manuscript received September 1, 1997; revised January 11, 1999. The material in this correspondence was presented in part at the IEEE International Symposium on Information Theory, Cambridge, MA, August [16][17][18][19][20][21]1998.…”