Electrical Engineering-Systems Dept.

*** SEMINAR ***

Ayal Hitron

(M.Sc. student under the supervision of Dr. Uri Erez)

on the subject:

Linear Coding for Gaussian Channels

We address the problem of communication over modulo-additive channels using linear codes over the ring Z_M. It has long been known that linear codes over finite fields achieve the capacity of any symmetric channel. Recently, there has been rising interest in structured codes for Gaussian network problems. In such problems, the codes need to be closed with respect to addition over the reals. Unfortunately, the addition in the finite field GF(p^m) does not translate to addition over the reals, except for the special case m=1. It is important to note that even for modulo-additive channels, whose capacity is achieved by a uniform input distribution,  linear codes with respect to addition over the reals do not always achieve the channel capacity. Our aim is to answer the question whether there exist capacity-achieving linear codes for the case of the modulo-additive Gaussian channel.

In recent works by Como and Fagnani, the capacity of linear codes, as well as their error exponent, were studied for codes over general Abelian groups. However, they gave only non-explicit expressions, so it remained an open question whether the capacity of the modulo-additive AWGN with M-PAM input can be achieved using linear codes (where the linearity is with respect to addition over the reals). In the present work we answer this question to the affirmative, in the case where M is a power of a prime. This also establishes that good lattices can be obtained via construction A from linear codes over Z_M. We also introduce an expurgation achievable error exponent for linear codes over modulo-additive channels, and discuss the tightness of this error exponent for small rates.