[Paper Review] Fountain Codes under Maximum Likelihood Decoding

Francisco Lázaro|arXiv (Cornell University)|Jun 27, 2017

Coding theory and cryptography1 references7 citations

TL;DR

This dissertation presents a comprehensive analysis of fountain codes—specifically LT and Raptor codes—under maximum likelihood (ML) decoding, focusing on inactivation decoding for improved efficiency. It introduces a dynamical programming approach to analyze decoding complexity and failure probability, derives tight upper bounds on decoding failure for q-ary Raptor codes, and proposes a new parallel concatenated fountain code design that achieves significantly lower failure rates than standard LT codes when using MDS precode, especially at moderate erasure rates.

ABSTRACT

This dissertation focuses on fountain codes under maximum likelihood (ML) decoding. First LT codes are considered under a practical and widely used ML decoding algorithm known as inactivation decoding. Different analysis techniques are presented to characterize the decoding complexity. Next an upper bound to the probability of decoding failure of Raptor codes under ML decoding is provided. Then, the distance properties of an ensemble of fixed-rate Raptor codes with linear random outer codes are analyzed. Finally, a novel class of fountain codes is presented, which consists of a parallel concatenation of a block code with a linear random fountain code.

Motivation & Objective

• To analyze the decoding complexity of LT codes under inactivation decoding using a dynamical programming approach.
• To derive the probability distribution of inactivations and develop an approximate analysis for code design.
• To establish a tight upper bound on the decoding failure probability of q-ary Raptor codes under ML decoding, leveraging the outer code's weight enumerator.
• To extend the analysis to Raptor codes with linear random outer codes and fixed-rate settings, characterizing their distance spectrum and minimum distance growth.
• To propose and analyze a new parallel concatenated fountain code architecture using an MDS precode to significantly reduce decoding failure probability.

Proposed method

• Employs a dynamical programming framework to model the expected number of inactivations in LT code decoding under inactivation decoding.
• Derives a binomial approximation for the inactivation count distribution and uses it to guide code design.
• Develops an upper bound on the decoding failure probability of q-ary Raptor codes by analyzing the weight enumerator of the outer code.
• Introduces a heuristic approximation for inactivation decoding in Raptor codes, enabling practical code design.
• Analyzes a Raptor code ensemble with linear random outer codes, deriving the average weight enumerator and its growth rate.
• Proposes a new parallel concatenated fountain code structure where a block code (especially MDS) is concatenated with a linear random fountain code to enhance performance.

Experimental results

Research questions

• RQ1 What is the expected number of inactivations in LT code decoding under inactivation decoding, and how can it be modeled using dynamical programming?
• RQ2 How does the probability distribution of inactivations behave, and can it be approximated for practical code design?
• RQ3 What is a tight upper bound on the decoding failure probability of q-ary Raptor codes under ML decoding, and how does it relate to the outer code's weight enumerator?
• RQ4 How can the performance of Raptor codes with linear random outer codes be characterized in terms of minimum distance growth?
• RQ5 Can a parallel concatenated fountain code design with an MDS precode achieve significantly lower failure probabilities than standard LT codes?

Key findings

• The dynamical programming approach provides an accurate analysis of the expected number of inactivations in LT code decoding, enabling precise code design.
• The probability distribution of inactivations is analytically characterized, and a lower-complexity binomial approximation is derived for practical use.
• A tight upper bound on the decoding failure probability of q-ary Raptor codes is established, which remains accurate even in the error floor region.
• For Raptor codes with linear random outer codes, the average weight enumerator and its growth rate are derived, and conditions for linear minimum distance growth with block length are provided.
• The proposed parallel concatenated fountain code with an MDS precode achieves failure probabilities several orders of magnitude lower than standard LRFC codes, especially at moderate erasure rates.
• The analysis demonstrates that the new code structure enables performance comparable to standard Raptor codes while offering significant gains in reliability under the same conditions.