D-IRA Codes Over Integer Rings for Lattice-Based Multiple Access

We study capacity-approaching irregular repeat accumulate (IRA) codes over integer rings <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{m}}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula>-PAM signaling, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m=1,2,\cdots $ </tex-math></inline-formula>. Such codes belong to the ensemble of lattice codes. First, the effect of zero-divisors on the iterative belief-propagation (BP) decoding is analyzed. We show that the symmetric Gaussian approximation is invalid for existing IRA codes over integer rings. Then we propose a new doubly IRA (D-IRA) ring code structure, which consists of irregular multiplier distribution and irregular node-degree distribution. Such structure can restore the symmetry and optimize the BP decoding threshold. Numerical result demonstrates the improved performance of D-IRA codes for AWGN channel. Also, it is demonstrated that the lattice based multiple-access scheme with D-IRA code performs within 3 dB the capacity limit.