[Paper Review] A Method to determine Partial Weight Enumerator for Linear Block Codes

Saïd Nouh, Bouchaib Aylaj+1 more|arXiv (Cornell University)|Dec 25, 2012

Advanced Wireless Communication Techniques36 references3 citations

TL;DR

This paper presents a novel method to compute the partial weight enumerator (PWE) of binary linear block codes using the error impulse technique and Monte Carlo simulation, enabling accurate upper bounds on soft decision maximum likelihood decoding error probability. The approach successfully determines previously unknown weight distributions for shortened BCH(130,66), BCH(103,47), and BCH(111,55) codes, derived from primitive BCH(255,191,17) and BCH(127,71,19) codes.

ABSTRACT

In this paper we present a fast and efficient method to find partial weight enumerator (PWE) for binary linear block codes by using the error impulse technique and Monte Carlo method. This PWE can be used to compute an upper bound of the error probability for the soft decision maximum likelihood decoder (MLD). As application of this method we give partial weight enumerators and analytical performances of the BCH(130,66), BCH(103,47) and BCH(111,55) shortened codes; the first code is obtained by shortening the binary primitive BCH (255,191,17) code and the two other codes are obtained by shortening the binary primitive BCH(127,71,19) code. The weight distributions of these three codes are unknown at our knowledge.

Motivation & Objective

• To develop an efficient computational method for determining the partial weight enumerator (PWE) of binary linear block codes when full weight distribution is unknown.
• To enable accurate upper bounding of the error probability for soft decision maximum likelihood decoding (MLD) in codes with unknown weight spectra.
• To apply the method to specific shortened BCH codes—BCH(130,66), BCH(103,47), and BCH(111,55)—whose weight distributions were previously unknown.
• To demonstrate the feasibility and accuracy of the proposed technique through analytical performance evaluation on practical code constructions.
• To provide a scalable and computationally efficient alternative to exhaustive weight enumeration for long or complex linear codes.

Proposed method

• The method employs the error impulse technique to systematically generate and analyze error patterns in the code space.
• Monte Carlo simulation is used to sample and estimate the distribution of codeword weights without enumerating all possible codewords.
• The partial weight enumerator is computed by tracking the frequency of codewords with weights up to a specified maximum, based on random sampling.
• The approach leverages the structure of shortened codes derived from known primitive BCH codes to reduce computational complexity.
• Error probability upper bounds are analytically derived from the computed PWE using standard union bound techniques in coding theory.
• The method is implemented iteratively, with convergence monitored by variance reduction in the Monte Carlo estimates.

Experimental results

Research questions

• RQ1 Can the partial weight enumerator of a binary linear block code be efficiently computed when the full weight distribution is intractable?
• RQ2 To what extent can the error impulse technique combined with Monte Carlo sampling improve the accuracy and speed of PWE estimation?
• RQ3 How accurately can the PWE estimate the upper bound on the soft decision MLD error probability for shortened BCH codes?
• RQ4 What is the performance of the method on specific shortened BCH codes like BCH(130,66), BCH(103,47), and BCH(111,55) with unknown weight distributions?
• RQ5 Can this method be generalized to other classes of linear block codes beyond shortened BCH codes?

Key findings

• The proposed method successfully computes the partial weight enumerator for BCH(130,66), BCH(103,47), and BCH(111,55) codes, which were previously unknown.
• The Monte Carlo-based estimation achieves stable and convergent results with a manageable number of samples, indicating computational feasibility.
• The method enables the derivation of tight upper bounds on the error probability for soft decision MLD, which are critical for system design.
• The weight distributions of the three shortened codes were not previously known, and this work provides the first analytical estimates.
• The error impulse technique effectively samples the relevant weight classes without requiring full enumeration, significantly reducing computational cost.
• The analytical performance results based on the PWE show consistent and reliable error probability bounds, validating the method’s accuracy.