Title

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

Authors

David C. Clark, Michigan Technological University
Vladimir Tonchev, Michigan Technological UniversityFollow

Document Type

Article

Publication Date

2013

Abstract

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.

Publisher's Statement

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

Publication Title

American Institute of Mathematical Sciences

Recommended Citation

Clark, D. C., & Tonchev, V. (2013). A new class of majority-logic decodable codes derived from polarity designs. American Institute of Mathematical Sciences, 7(2), 175-186. http://doi.org/10.3934/amc.2013.7.175
Retrieved from: https://digitalcommons.mtu.edu/math-fp/88