All the codeword symbols in polar codes have the same SER under the SC decoder

TL;DR

This paper proves that in polar codes constructed with a specific kernel over finite fields, all codeword symbols experience identical symbol error rates under the SC decoder, regardless of code parameters.

Contribution

It establishes the uniform SER property for all codeword symbols in polar codes with the given kernel over finite fields, extending understanding of error performance.

Findings

All codeword symbols have the same SER under SC decoding.

The result holds for any code length and dimension.

The proof applies to channels over finite fields with specific algebraic properties.

Abstract

We consider polar codes constructed from the $2\times 2$ kernel $\begin{bmatrix} 1 & 0 \\ \alpha & 1 \end{bmatrix}$ over a finite field $\mathbb{F}_{q}$, where $q=p^s$ is a power of a prime number $p$, and $\alpha$ satisfies that $\mathbb{F}_{p}(\alpha) = \mathbb{F}_{q}$. We prove that for any $\mathbb{F}_{q}$-symmetric memoryless channel, any code length, and any code dimension, all the codeword symbols in such polar codes have the same symbol error rate (SER) under the successive cancellation (SC) decoder.