Folded tree maximum-likelihood decoder for Kronecker product-based codes

In this paper, we propose efficient maximum-likelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product G
_n
= F
^⊗n
of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed- Muller codes are well known examples of such KPB codes. The triangular structure of G
_n
enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of G
_n
and the “tree folding” technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κ-folding operation transforms the binary tree with N levels into a non-binary tree with N=2
^κ
levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find (n over κ) different folding which lead to decoders with different complexity, for a given code. Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.