System and method for low-density parity check (LDPC) code design

A system for determining a low-density parity check (LDPC) code design for a half-duplex relay channel comprising:

• a variable node decoder configured to calculate variable node density functions ƒV(1)(i) and ƒV(2)(i) within a first codeword region and a second codeword region, respectively, said variable node density functions ƒV(1)(i) and ƒV(2)(i) being density functions of output edges of variable nodes with degree i within said first codeword region and said second codeword region, respectively;
• a check node decoder configured to calculate check node density functions ƒC(1)(j) and ƒC(2)(j) within said first codeword region and said second codeword region, respectively, said check node density functions ƒC(1)(j) and ƒC(2)(j) being density functions of output edges of check nodes with degree j within said first codeword region and said second codeword region, respectively;
• a processor configured to calculate mixture density functions of output check edges connecting variable nodes in said first codeword region and said second codeword region to determine an LDPC code design,
• wherein, if a signal to noise ratio (SNR) of said first codeword region is different than a signal to noise ratio of said second codeword region, then
• ƒV(1)(i)≠ƒV(2)(i) for each i and ƒC(1)(j)≠ƒC(2)(j) for each j.