On the Equivalence between Pre-transformed and Parity-check Monomial Codes

Polar codes and Reed-Muller codes belong to a family of codes called monomial codes. In this work, we study pre-transformed monomial codes, which cover several constructions including parity-check (PC) codes and PAC codes. We show that any pre-transformed monomial code can be transformed into a parity-check monomial code with the same codewords, and give an explicit algorithm for this transformation. We further prove that for certain monomial codes, the minimum weight is invariant under pre-transformation, but specific pre-transformation matrices can be constructed to reduce the number of minimum-weight codewords. These results offer theoretical support for the success of various heuristics, e.g., PAC codes attain dispersion bound, and provide guidance for designing short codes.