Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Natarajan, Lakshmi Prasad and Krishnan, Prasad (2022) Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity. In: 2022 IEEE International Symposium on Information Theory, ISIT 2022, 26 June 2022 through 1 July 2022, Espoo.

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Abstract

We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as Cn(r,m), and their duals are denoted as Bn(r,m). The length of these codes is nm, where n ≥ 2, and r is their 'order'. When n = 2, Cn(r,m) is the RM code of order r and length 2m. The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to Bn(r,m) as the Berman code and Cn(r,m) as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al. (2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. © 2022 IEEE.

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IITH Creators:
IITH CreatorsORCiD
Natarajan, Lakshmi Prasadhttp://orcid.org/0000-0003-1552-5240
Item Type: Conference or Workshop Item (Paper)
Additional Information: The work of Lakshmi Prasad Natarajan was supported by SERB-DST via grant MTR/2019/001454. Prasad Krishnan acknowledges support from SERBDST project CRG/2019/005572.
Uncontrolled Keywords: Automorphism groups; Automorphisms; Binary erasure channel; Channel's capacity; Generalisation; MAP decoding; Odd prime; Reed-Muller codes
Subjects: Electrical Engineering
Divisions: Department of Electrical Engineering
Depositing User: . LibTrainee 2021
Date Deposited: 01 Sep 2022 10:57
Last Modified: 01 Sep 2022 10:57
URI: http://raiith.iith.ac.in/id/eprint/10382
Publisher URL: http://doi.org/10.1109/ISIT50566.2022.9834598
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