Elsevier

Signal Processing

Volume 83, Issue 7, July 2003, Pages 1467-1486
Signal Processing

Robust indexing of lattices and permutation codes over binary symmetric channels

https://doi.org/10.1016/S0165-1684(03)00063-XGet rights and content

Abstract

In this paper we develop two families of indexing methods for permutation codes and lattices, the lexicographical and binomial indexing families. The parameters of a method within a family can be optimized with respect to a given criterion in order to obtain a good indexing method on the considered set of vectors to be indexed. The proposed indexing methods can be straightforwardly applied for error resilient coding, but also to other applications as e.g. the magnetic recording which is based on (d,k) constrained codes. We have tested the performance of the proposed method at the optimization of the indexing method of a codebook with respect to the channel distortion for several types of lattices and other structured vector quantizers over binary symmetric channels.

Introduction

The enumeration of lattice codevectors has well established solutions, as several enumeration techniques for the common used lattices and lattice truncations exist [4], [8], [10] but the error resilience of such indexing techniques remains an open research issue. The work of Hung et al. [7] was one of the first to consider this problem from the point of view of the channel robustness of a lattice indexing technique, but their work was mainly restricted to Zn lattices. In [14] we have studied several indexing techniques for truncated lattices and for multiple-scale lattice vector quantizer (VQ) structures [13]. However, in [14], we have only presented several indexing methods and we have chosen that giving the best error resilience, without further attempts to modify the indexing in order to optimize the performance. In the present study we introduce a parameterization of the indexing methods, allowing thus further optimization for a better robustness over a noisy channel. The methods we develop here can be used for optimizing the indexing in several applications: permutation codes, lattice vector quantizers and multiple scale lattice vector quantizers. The criterion with respect to which the optimization is carried on, is not restricted to be only the channel distortion and is generally given by the practical application.

The core of this work is the enumeration of an unsigned leader class obtained through all the permutations of the components of a leader vector.

The main contributions of the study are presented in the 3 Indexing of an unsigned leader class, 5 Indexing optimization which consist of the description of two families of indexing methods for an unsigned leader class and the optimization procedures inside these families. In Section 4 we consider the case of a signed leader class. Section 6 is devoted to some theoretical aspects concerning the average channel distortion of a codebook over a binary symmetric channel. The remaining sections present the results obtained with the proposed indexing methods for the permutation codes, lattice vector quantizers and multiple scale lattice vector quantizers.

Section snippets

Leader class and lattices

A leader vector v is a vector which has non-negative elements ordered decreasingly. Every such vector defines a set of vectors dubbed as the leader class, containing all vectors obtained by a signed permutation of v, described in more detail by the following two properties:

  • (1)

    Together with v, all the vectors obtained by permutations of entries in v form the unsigned leader class of v.

  • (2)

    The vectors obtained through permutations and (constrained) sign switching form the leader class of the vector v.

Indexing of an unsigned leader class

We propose here two types of indexing methods for the codevectors in an unsigned leader class: the generalized lexicographical indexing and the generalized binomial indexing. We will further see that these methods can be unified as a more general indexing technique. In all that follows we will consider the leader vector v=[v0vn−1] having m possible values for its entries vk∈{i0,i1,…,im−1}⊂Nm,k=0,…,n−1. The unsigned leader class of a leader vector v is symbolized by L(v). For notational

Leader class with null parity

A zero value parity for a leader class means that there is no constraint on the number of either positive or negative components in a vector from that class. A very simple indexing method for the vector of signs can be realized by assigning to each strictly negative component the value 1 and to each strictly positive component the value 0, the resulting binary string being the index IS.

Leader class with non-null parity

If a leader class has a non-null parity, α, 1 or −1, then its vectors should have only an even or odd number

Optimization of the lexicographical indexing

We have seen in Section 3.3.1 that the generalized lexicographical indexing is described using a quasi order relation for the possible values of the components for each dimension. If the quasi order relation is the same for all the dimensions, we obtain the classical lexicographical indexing.

The optimization of the indexing procedure in the context of the tree from Fig. 1 is in fact the choice of a quasi order relation for each dimension, such that a given error criterion should be minimized.

Average channel distortion

The average channel distortion for all the variants of indexing a codebook could be an indicator of how good a particular indexing is or, at least, if it is below or over the average. This is why we calculate in this section the average channel distortion over a binary symmetric channel for a given codebook.

Due to Shannon's separation theorem one may consider independently the channel distortion provided the source distortion is optimal. However, if this is not the case, all the procedures that

Indexing of permutation codes

As mentioned before the definition of an unsigned leader class or of a leader class matches the definitions of the variant I and variant II permutation codes. Therefore the indexing methods of leader classes apply straightforwardly to permutation codes. We consider here the variant II of permutation codes corresponding to the signed leader class.

Indexing of lattice codevectors

We present here two methods for indexing the vectors of a truncated lattice. In the first one the vectors from the same leader class are indexed in a given contiguous interval of integer numbers, whilst in the second the vectors from different leader classes are mixed together. We denote the first method by “separate leaders” (SLd) and the second one by “mixed leaders” (MLd).

Indexing of MSLLVQ

Passing from a truncated lattice to a multiple scale structure is done in a similar manner to passing from a single leader class to a truncated lattice. The multiple scale structures are specified as ini×li, where li is the last leader class that is contained in the truncation and ni indicates how many times the corresponding truncation is considered in the codebook (having assigned, of course, different scales). For instance 2×12+7 denotes a quantization structure containing three

Conclusions

This study proposed and analyzed new indexing of codes with regular structure: permutation codes, truncated lattices and multiple-scale lattice structures. We parametrized three types of indexing methods, generalized lexicographical and generalized binomial (two variants) in order to allow optimization with respect to various criteria. There are two methods for optimizing these indexing methods: an iterative method and a greedy method. The greedy optimization method gives better results than

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This work has been supported by Academy of Finland, project No. 44876 (Finnish Centre of Excellence Program (2000-2005)).

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