Lower bounds on the state complexity of linear tail-biting trellises are presented. One bound generalizes the total-span bound, while another bound can be regarded as a generalization of the cut-set bound. It is shown by examples that the new bounds may be tighter than any of the existing lower bounds.

We give a new definition of generalized Hamming weights of nonlinear codes and a new interpretation connected with it. These generalized weights are determined by the entropy/length profile of the code. We show that this definition characterizes the performance of nonlinear codes on the wire-tap channel of type II. The new definition is invariant under translates of the code, it satisfies the property of strict monotonicity and the generalized Singleton bound. We check the relations between the generalized weight hierarchies of Z/sub 4/-linear codes and their binary image under the Gray map. We also show that the binary image of a Z/sub 4/-linear code is a symmetric, not necessarily rectangular code. Moreover, if this binary image is a linear code then it admits a twisted squaring construction.

Weighted entropy profiles and a new bound, the weighted coordinates bound, on the state complexity profile of block codes are presented. These profiles and bound generalize the notion of dimension/length profile (DLP) and entropy/length profile (ELF) to block codes whose symbols are not drawn from a common alphabet set, and in particular, group codes. Likewise, the new bound may improve upon the DLP and ELF bounds fur linear and nonlinear block codes over fields. However, it seems that the major contribution of the proposed bound is to the study of trellis complexity of block codes whose different coordinates are drawn from different alphabet sets. The label code of lattice and nonlattice periodic packings usually has this property. The construction of a trellis diagram for a lattice and some related bounds are generalized to periodic packings by introducing the fundamental module of the packing, and using the new bound on the state complexity profile. This generalization is limited to a given coordinate system. We show that any bounds on the trellis structure of block codes, and in particular, the bound presented in this work, are applicable to periodic packings.

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 well-known 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.

In this correspondence, the trellis representation of the Kerdock and Delsarte-Goethals codes is addressed. It is shown that the states of a trellis representation of DG(m,/spl delta/) under any bit-order are either strict-sense nonmerging or strict-sense nonexpanding, except, maybe, at indices within the code's distance set. For /spl delta//spl ges/3 and for m/spl ges/6, the state complexity, s/sub max/[DG(m,/spl delta/)], is found. For all values of m and /spl delta/, a formula for the number of states and branches of the biproper trellis diagram of DG(m, /spl delta/) is given for some of the indices, and upper and lower bounds are given for the remaining indices. The formula and the bounds refer to the Delsarte-Goethals codes when arranged in the standard bit-order.

The Barankin (1949) bound is a realizable lower bound on the mean-square error (MSE) of any unbiased estimator of a (nonrandom) parameter vector. We present a Barankin-type bound which is useful in problems where there is a prior knowledge on some of the parameters to be estimated. That is, the parameter vector is a hybrid vector in the sense that some of its entries are deterministic while other are random variables. We present a simple expression for a positive-definite matrix which provides bounds on the covariance of any unbiased estimator of the nonrandom parameters and an estimator of the random parameters, simultaneously. We show that the Barankin bound for deterministic parameters estimation and the Bobrovsky-Zakai (1976) bound for random parameters estimation are special cases of our proposed bound.

We use the term hybrid parameter vector to refer to a vector which consists of both random and non-random parameters. We present a novel Barankin-type lower bound which bounds the estimation error of a hybrid parameter vector. The bound is expressed in a simple matrix form which consists of a non-Bayesian bound on the non-random parameters, a Bayesian bound on the random parameters, and the cross terms. We show that the non-Bayesian Barankin (1949) bound for deterministic parameters estimation and the Bobrovsky-Zakai (1976) Bayesian bound for random parameters estimation are special cases of the new bound. Also, the multi-parameter Cramer-Rao bound, in its Bayesian or non-Bayesian versions, are shown to be special cases of the new bound.

We report results of a research in which we studied the problem of determining the threshold signal to noise ratio (SNR) between large and small errors in the estimation of the direction of arrival (DOA) of a radiating, far-field source in the presence of another. Using the Barankin lower bound (BB) we examine the conditions under which achievable mean square error (MSE) performance of any unbiased DOA estimator deviates substantially from the Carmer-Rao lower bound (CRB). We present expressions for the threshold SNR as a function of the source-array geometry and the sources SNR where one and two sources, of known/unknown spectral parameters and DOAs, are present.