A NEW CLASS OF MAJORITY-LOGIC DECODABLE CODES DERIVED FROM POLARITY DESIGNS
The polarity designs, introduced in [9], are combinatorial 2-designs having the same parameters as a projective geometry design PG(s)(2s, q) formed by the s-subspaces of PG(2s, q), s >= 2,q - p(t), p prime. If q - p is a prime, a polarity design has also the same p-rank as PG(s) (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 PG(s) (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 2(2s+1) and order s.
majority-logic decoding polarity designs reed-muller codes hamada's conjecture