Extended Binary Chirps Codebooks for Non-Coherent Communications

Binary chirps (BCs) are exponentiated 2nd-order Reed-Muller codes, which have interesting geometric and algebraic features, one of which is the close connection to the diagonal part of the Clifford group, which is the 2nd level of the Clifford hierarchy. We develop a novel transvection based method to analyze the diagonal Clifford hierarchy. Using this, we identify a connection of recently proposed generalized BCs with the 3rd level of Clifford hierarchy. Then, we propose two systematic extensions of the BC codebook to an arbitrary Clifford hierarchy level and find their minimum distances. In these extensions, the number of codewords grows exponentially with the hierarchy level. For decoding, we design a low-complexity decoding approach for the extended BCs, using the Howard algorithm for BC decoding as a component. Through simulations, we show that the performance of the proposed low-complexity decoder can achieve performance very close to the exhaustive search with significantly reduced complexity.