{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LPDecoding_allerton03

LPDecoding_allerton03 - LP Decoding Jon Feldman Industrial...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
LP Decoding Jon Feldman * Industrial Engineering and Operations Research Columbia University, New York, NY, 10027 [email protected] David R. Karger Laboratory for Computer Science MIT, Cambridge, MA, 02139 [email protected] Martin J. Wainwright Electrical Engineering and Computer Science UC Berkeley, CA, 94720 [email protected] Abstract. Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximum-likelihood (ML) decoding. An arbitrary binary-input memoryless channel is considered. This treatment of the LP decoding method places our previous work on turbo codes [6] and low-density parity-check (LDPC) codes [8] into a generic framework. We define the notion of a proper relaxation, and show that any LP decoder that uses a proper relaxation exhibits many useful properties. We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels. The fractional distance of an LP decoder is defined, and it is shown that LP decoders correct a number of errors equal to half the fractional distance. We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code. We show that if a relaxation is symmetric, one may assume that the all-zeros codeword is transmitted. 1 Introduction The problem of maximum-likelihood (ML) decoding is to find the codeword most likely to have been transmitted, given a corrupted codeword from a noisy channel. Linear programming is the problem of finding an optimal solution to a system of linear inequalities under a linear objective function [2]. In this paper, we consider linear programming (LP) formulations of the ML decoding problem on binary codes. We use LP variables to represent code bits, and the LP objective function is defined by the channel likelihood ratios. Previous work on LP decoding [6, 4, 7, 8, 5] has focused on two specific cases: turbo codes [1] and low-density parity-check codes [11]. These two families of codes have received a lot of attention recently due to their excellent performance. Performance bounds for LP decoding in these cases are for specific LP formulations, code constructions, and/or channel models. In this paper we consider LP decoders for arbitrary binary codes, under an arbitrary binary- input memoryless channel. We provide a framework for designing LP decoders, and general techniques for analyzing them. Central to every LP decoder is its associated polytope : the set of points that satisfy the constraints of the LP. A decoding polytope should contain ev- ery codeword, and should also exclude every binary word that is not a codeword. We define such polytopes as proper . We show that LP decoders that use proper polytopes have the ML certificate property: whenever they output a codeword, it is guaranteed to be the ML codeword.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern