Irina Bocharova

Irina Bocharova, Maiara F. Bollauf, Boris Kudryashov

A new geometric shaping technique, referred to as coset shaping, is proposed and analyzed for coded QAM and PAM signaling. This method can be applied to both information and parity bits without introducing additional complexity. It is shown that, as the error-correcting code length and the modulation order grow, the gap to capacity of the proposed shaping scheme can be made arbitrarily small. A Gallager-type bound is provided together with numerical results, including performance comparisons for the shaping scheme combined with short and mid-length binary-coded, as well as nonbinary LDPC-coded QAM signaling

Irina Bocharova, Boris Kudryashov, Henk D. L. Hollmann et al.

In modern data storage systems, non-binary LDPC codes for recovering from disk failures are increasingly considered strong competitors to MDS codes such as Reed-Solomon codes. Since disk failures can be modeled as erasures, we analyze non-binary LDPC codes over a $q$-ary field in the $q$-ary erasure channel, relative to MDS codes. Our focus is on non-binary LDPC codes whose parity-check matrix is obtained by replacing the non-zero entries of a binary base matrix by elements of a $q$-ary finite field. For such LDPC codes, we introduce the notion of ultimate distance, which upper-bounds their minimum distance. We derive a random-coding bound on the number of non-correctable erasure patterns for the Gallager ensemble of regular non-binary LDPC codes under maximum-likelihood decoding. An algorithm for finding the ultimate distance is presented. A low-complexity algorithm for searching for the minimum distance of the non-binary LDPC code is proposed. Finally, we construct examples of non-binary LDPC codes achieving the ultimate distance.