Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2q)

Authors:
Amir J. Salomon;Ofer Amrani
Affiliations:
Department of Electrical Engineering---Systems, School of Electrical Engineering, Tel-Aviv University, Tel-Aviv, Israel;Department of Electrical Engineering---Systems, School of Electrical Engineering, Tel-Aviv University, Tel-Aviv, Israel
Venue:
Designs, Codes and Cryptography
Year:
2007

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Abstract

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall 驴(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 q ) and set partitioning based on partition chains of length-2 l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.