Majority Logic Decoding With Subspace Designs

Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as codewords. Peterson and Weldon (1972) extended Rudolph's algorithm to a two-step majority logic decoder correcting the same number of errors as Reed's celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved. For a known infinite series of subspace designs the reduction of complexity is exponential.