A projection decoding of a binary extremal self-dual code of length 40

Abstract

As far as we know, there is no decoding algorithm of any binary self-dual [40, 20, 8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40, 20, 8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two new efficient decoding algorithms for an extremal binary doubly-even self-dual [40, 20, 8] code $$C_{40,1}^{DE}$$C40,1DE by hand with the help of a Hermitian self-dual [10, 5, 4] code $$E_{10}$$E10 over GF(4). The main idea of this decoding is to project codewords of $$C_{40,1}^{DE}$$C40,1DE onto $$E_{10}$$E10 so that it reduces the complexity of the decoding of $$C_{40,1}^{DE}$$C40,1DE. The first algorithm is called the representation decoding algorithm. It is based on the pattern of codewords of $$E_{10}$$E10. Using certain automorphisms of $$E_{10}$$E10, we show that only eight types of codewords of $$E_{10}$$E10 can produce all the codewords of $$E_{10}$$E10. The second algorithm is called the syndrome decoding algorithm based on $$E_{10}$$E10. It first solves the syndrome equation in $$E_{10}$$E10 and finds a corresponding binary codeword of $$C_{40,1}^{DE}$$C40,1DE.