Low Density Parity Check Codes

1. Low Density Parity Check Codes LDPC ( Low Density Parity Check ) codes are a class of linear bock code. The term “Low Density” refers to the characteristic of the parity check matrix which contains only few ‘1’s in comparison to ‘0’s. LDPC codes are arguably the best error correction codes in existence at present. LDPC codes were first introduced by R. Galager in his PhD thesis in 1960 and soon forgotten due to introduction of Reed-Solomon codes and the implementation issues with limited technological knowhow at that time. The LDPC codes were rediscovered in mid 90s by R. Neal and D. Mackay at the Cambridge University. We can define N bit long LDPC code in terms of M number of parity check equations and describing those parity check equations with a M x N parity check matrix H. Where, M – Number of parity check equations N – Number of bits in the codeword

2. Row weight - number of ‘1’s in a row Number of Number of symbols taking part in a parity check and changes, therefore this is an irregular parity check matrix Column weight - number of ‘1’s in a column Number of times a symbol taking part in parity checks Low Density Parity Check Codes Consider the 6 bit long codeword in the form which satisfies 3 parity check equations as shown below. We can now define 3x6 parity check matrix as, The density of ‘1’s in LDPC code parity check matrix is very low If the parity check matrix has uniform row weight and uniform column weight (same number of ‘1’ in a column and same number of ‘1’ in a row) we call that a regular parity check matrix.

3. The parity checkmatrixdefines a rate , code where We can generate the codeword in by multiplying message with generator matrix Low Density Parity Check Codes Codeword is said to be valid if it satisfies the syndrome calculation We can obtain the generator matrix from parity check matrix by, 1.) Arranging the parity check matrix in systematic form using row and column operations 2.) Rearranging the systematic parity check matrix 3.) We can verify our results as

4. Low Density Parity Check Codes Tanner graph is a graphical representation of parity check matrix specifying parity check equations. Tanner graph consists of N number of variable nodes and M number of check nodes In Tanner graph mth check node is connected to nth variable node if and only if nth element in mth row in parity check matrix is a ‘1’. The marked path z2 → c1 → z3 → c6 → z2 is an example for short cycle of 4 The number of steps needed to return to the original position is known as the girth of the code

5. Suppose we have codeword as follows: • where each is either ‘0’ or ‘1’ and codeword now has three parity-check • equations. Low Density Parity Check Codes a.) Determine the parity check matrix H by using the above equation b.) Show the systematic form of H by applying Gauss Jordan elimination c.) Determine Generator matrix Gfrom Hand prove G * HT = 0 d.) Find out the dimension of the H, G e.) State whether the matrix is regular or irregular

6. Convert into systematic form using basic row and operations (try to avoid column operations) Low Density Parity Check Codes 6 columns and 3 rows

7. Low Density Parity Check Codes Dimensions of H = 3x6 Dimensions of G = 3x6 Since the column weight and row weight changes, this is an irregular parity check matrix

8. Low Density Parity Check Codes 2. The parity check matrix H of LDPC code is given below: a.) Determine the degree of rows and column b.) State whether the LDPC code is regular or irregular c.) Determine the rate of the LDPC code d.) Draw the tanner graph representation of this LDPC code. e.) What would be the code rate if we make rows equals to column f.) Write down the parity check equation of the LDPC code

9. Row weight Column weight Low Density Parity Check Codes Dimensions of the matrix = 6x12 Since the parity check matrix has a uniform row weight and uniform column weight, this is a regular LDPC parity check matrix Since the parity check matrix has a uniform row weight and uniform column weight, this is a regular LDPC parity check matrix Code rate If we make the number of rows equal to columns code rate will be equal to ‘0’

10. Low Density Parity Check Codes The parity check equations of the matrix are

11. Dimensions of the matrix = 3x6 Low Density Parity Check Codes 3. Consider parity check matrix H generated in question 1, a.) Determine message bits length K, parity bits length M, codeword length N b.) Use the generator matrix G obtained in question 1 to generate all possible codewords c. All possible message set All possible codeword set

12. Check Nodes Variable Nodes Low Density Parity Check Codes Question 4. a.) What is the difference between regular and irregular LDPC codes? b.) What is the importance of cycles in parity check matrix? c.) Identify the cycles of 4 in the following tanner graph. If the parity check matrix has uniform row weight and uniform column weight (same number of ‘1’ in a column and same number of ‘1’ in a row) we call that a regular parity check matrix. The decoding algorithm assumes that the LDPC code is cycle free ( large girth sizes ). The short cycles in the code ( cycles with a girth of 4 and cycles of girth of 6 ) weakens the code. Therefore the codes must be carefully constructed to be free of short cycles.