A new class of majority-logic decodable codes derived from polarity designs

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The polarity designs, introduced in [9], are combinatorial 2-designs having the same parameters as a projective geometry design PGs(2s,q) formed by the s -subspaces of PG(2s,q), s ≥ 2 , q = pt , p prime. If q = p is a prime, a polarity design has also the same p -rank as PGs(2s,p) . If q = 2 , any polarity 2-design is extendable to a 3-design having the same parameters and 2-rank as an affine geometry design AG s+1(2s + 1,2) formed by the ( s + 1 ) -subspaces of AG(2s + 1,2). It is shown in this paper that a linear code being the null space of the incidence matrix of a polarity design can correct by majority-logic decoding the same number of errors as the projective geometry code based on PGs(2s,q) . In the binary case, any polarity 3-design yields a binary self-dual code with the same parameters, minimum distance, and correcting the same number of errors by majority-logic decoding as the Reed-Muller code of length 22s + 1 and order s.

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Copyright © 2013 American Institute of Mathematical Sciences. Publisher’s version of record: https://dx.doi.org/10.3934/amc.2013.7.175

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American Institute of Mathematical Sciences