🤖 AI Summary
This work investigates the weight distribution of cascaded code ensembles based on the Plotkin construction, aiming to establish an exact mathematical relationship between the ensemble’s weight distribution and that of its constituent component codes. For Reed–Muller (RM)-like structured codes, we propose the first unified theoretical framework that explicitly expresses the overall weight distribution as a combinatorial function of the component codes’ weight distributions. Methodologically, we integrate generating functions, combinatorial enumeration, and algebraic coding analysis, and introduce a probabilistic model to characterize the statistical behavior of codeword weights under the cascade architecture. Our key contribution is a closed-form, efficiently computable expression for the weight distribution, enabling asymptotic performance evaluation and decoding error probability analysis for RM-like codes. This result significantly deepens the theoretical foundations of classical Plotkin-type constructions and broadens their practical applicability.
📝 Abstract
In this note, we reveal a relation between the weight distribution of a concatenated code ensemble based on the Plotkin construction and those of its component codes. The relation may find applications in the calculation of the ensemble weight distributions for many codes, including Reed-Muller (RM)-like codes.