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Low Density Parity Check Codes An Introduction

Low Density Parity Check Codes An Introduction. Yuta Toriyama. yuta@ee.ucla.edu August 24, 2012. Channel Coding / FEC. Technique for controlling errors in transmission of data Redundancy in error-correcting code Hamming. Simple Parity Check Example. Simple scheme: repeat each bit 3 times

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Low Density Parity Check Codes An Introduction

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  1. Low Density Parity Check CodesAn Introduction Yuta Toriyama yuta@ee.ucla.edu August 24, 2012

  2. Channel Coding / FEC • Technique for controlling errors in transmission of data • Redundancy in error-correcting code • Hamming

  3. Simple Parity Check Example • Simple scheme: repeat each bit 3 times • Majority rule to recover single bit • (3,1) code, single error detection & correction 0 0 0 0 1 1 1 1

  4. Simple Parity Check Example • “Hypercube” representing gray code

  5. Simple Parity Check Example • “Hypercube” representing gray code

  6. Linear Block Codes • Generator matrix G (n x m) • Parity-check matrix H (m x k) • Generator matrix is a transformation from n to m dimensions • Codeword c is the nullspace (kernel) of the parity-check matrix • Example: Hamming(7,4)

  7. Low Density Parity Check Codes • Parity-check matrix is sparse • A particular LDPC code can be represented by a sparse bipartite graph (“Tanner graph”) • H is the bi-adjacency matrix of the Tanner graph

  8. Iterative Message Passing Algorithm • Iterative decoding algorithm • v-nodes and c-nodes pass “messages” to each other • Belief Propagation

  9. Non Binary LDPC • Numbers are limited elements in GF(2p) • Decoding is much more complicated • Performs better than binary LDPC, especially in the case of short or medium codeword lengths Davey, M.C.; MacKay, D.; , "Low-density parity check codes over GF(q)," Communications Letters, IEEE , vol.2, no.6, pp.165-167, June 1998

  10. Summary • LDPC codes show strong potential as a set of codes with very low BER • Computational complexity of decoding algorithms needs to be resolved to make NB-LDPC practical as well as allow for further constructive research

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