The Exponent of a Polarizing Matrix Constructed from the Kronecker Product

TL;DR

This paper analyzes how the exponent of a polarizing matrix, constructed via Kronecker product, relates to its components, providing formulas that aid in designing matrices with high polarization efficiency.

Contribution

It proves that the partial distances of a Kronecker product polarizing matrix are products of component distances and that its exponent is a weighted sum of component exponents.

Findings

Partial distances of the Kronecker product matrix are products of component distances.

The exponent of the Kronecker product matrix is a weighted sum of component exponents.

These formulas facilitate the design of large polarizing matrices with high exponents.

Abstract

The asymptotic performance of a polar code under successive cancellation decoding is determined by the exponent of its polarizing matrix. We first prove that the partial distances of a polarizing matrix constructed from the Kronecker product are simply expressed as a product of those of its component matrices. We then show that the exponent of the polarizing matrix is shown to be a weighted sum of the exponents of its component matrices. These results may be employed in the design of a large polarizing matrix with high exponent.