On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords

TL;DR

This paper derives a closed-form formula for enumerating polar codewords with weight 1.5 times the minimum weight, advancing understanding of their weight distribution beyond the minimum weight.

Contribution

It introduces a novel closed-form expression for codewords with weight 1.5 times the minimum weight in polar codes, based on algebraic structures and Minkowski sums.

Findings

Closed-form enumeration for weight 1.5 extit{d} codewords.

Potential extension of the method to higher weights.

Enhanced understanding of polar code weight distribution.

Abstract

The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.