On the Optimal Decomposition of the U-UV Codes

IEEE Access. 2026. Vol. 14. P. 50549–50557.

Kuvshinov A., Fominykh A., Ivanov F.

The recursive (U|U+V) construction, a generalization of which includes polar codes, provides a powerful framework for building complex codes from simpler components. However, existing approaches predominantly rely on fixed or symmetric tree architectures, overlooking the critical impact of decomposition choice on code performance. This paper addresses the challenge of optimal tree decomposition selection by presenting a framework for designing and analyzing multilevel (U|U+V) codes. Unlike previous approaches that focus on fixed architectures, our method systematically explores the entire design space of possible U-UV tree structures to identify constructions that optimally balance performance and complexity. The proposed approach evaluates each candidate decomposition by computing both the theoretical decoding error probability and computational complexity. Error probability analysis assumes that individual (U|U+V) components at the leaf nodes operate at the finite-blocklength Polyanskiy bound, while the complete construction employs sequential decoding. Computational complexity is quantified in terms of the total number of maximum-likelihood (ML) decoding operations required across all leaf nodes. By exhaustively comparing all feasible tree structures, our method enables the selection of optimal decompositions that minimize error probability under specific complexity constraints, providing a systematic design methodology for U-UV codes in practical communication systems.

Keywords: Concatenated codes, channel coding, polar codes, Gaussian channels