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- W4289655172 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 ${mathcal{C}_n}(r,m)$, and their duals are denoted as ${{mathcal{B}}_n}(r,m)$. The length of these codes is n <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>m</sup> , where n ≥ 2, and r is their ‘order’. When n = 2, ${mathcal{C}_n}(r,m)$ is the RM code of order r and length 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>m</sup> . 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 ${{mathcal{B}}_n}(r,m)$ as the Berman code and ${mathcal{C}_n}(r,m)$ 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." @default.
- W4289655172 created "2022-08-03" @default.
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- W4289655172 date "2022-06-26" @default.
- W4289655172 modified "2023-10-16" @default.
- W4289655172 title "Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity" @default.
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- W4289655172 doi "https://doi.org/10.1109/isit50566.2022.9834598" @default.
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