FR1.R3.3

Weight Structure of Low/High-Rate Polar Codes and Its Applications

Mohammad Rowshan, University of New South Wales (UNSW), Australia; Vlad-Florin Dragoi, Aurel Vlaicu University, Romania; Jinhong Yuan, University of New South Wales (UNSW), Australia

Session:
Polar codes 3

Track:
2: Modern Coding Theory

Location:
Ypsilon IV-V-VI

Presentation Time:
Fri, 12 Jul, 10:25 - 10:45

Session Chair:
Hsin-Po Wang,
Abstract
The structure of a linear block code is pivotal in defining fundamental properties, particularly the weight distribution. In this study, we characterize the Type II structure of polar codewords with weights less than twice the minimum weight $w_{min}$, utilizing the lower triangular affine (LTA) transform. We present a closed-form formula for their enumeration. Leveraging this structure and additionally characterizing the structure of weight $2w_{min}$, we ascertain the complete weight distribution of low-rate and, through the utilization of dual codes properties, high-rate polar codes, subcodes of Reed--Muller (RM) codes, and RMxPolar codes. Furthermore, we introduce a partial order based on the weight distribution and briefly explore its properties and applications in code construction and analysis.
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