Computing the Low-Weight codewords of Punctured and Shortened Pre-Transformed polar Codes

TL;DR

A deterministic algorithm to count the low-weight codewords of punctured and shortened pure and pre-transformed polar codes, regardless of the frozen bit set, puncturing/shortening pattern, or pretransformation is presented.

Abstract

In this paper, we present a deterministic algorithm to count the low-weight codewords of punctured and shortened pure and pre-transformed polar codes. The method first evaluates the weight properties of punctured/shortened polar cosets. Then, a method that discards the cosets that have no impact on the computation of the low-weight codewords is introduced. A key advantage of this method is its applicability, regardless of the frozen bit set, puncturing/shortening pattern, or pretransformation. Results confirm the method's efficiency while showing reduced computational complexity compared to stateof-the-art algorithms.

Figures (4)

• Figure 1:
• Figure 2:
• Figure 3: Transformation matrix for $N =8$ and $P = 4$
• Figure 4:

Theorems & Definitions (2)

• Example 3.1
• Example 4.1