# Majority-logic Decoding with Subspace Designs

## TL;DR

This paper explores majority-logic decoding for codes derived from subspace designs, showing they match the error correction capabilities of geometric design codes but with potentially much simpler decoding complexity.

## Contribution

It introduces the use of subspace designs for majority-logic decoding, demonstrating comparable error correction with reduced decoding complexity.

## Key findings

- Codes from subspace designs have the same decoding capability as geometric design codes.
- Decoding complexity can be significantly reduced using subspace design codes.
- The approach extends Rudolph's and Peterson-Weldon's decoding methods.

## Abstract

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 Rudolphs algorithm to a two-step majority logic decoder correcting the same number of errors than 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.