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

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 CDE 40,1 by hand with the help of a Hermitian self-dual [10, 5, 4] code E10 over GF (4). The main idea of this decoding is to project codewords of CDE 40,1 onto E10 so that it reduces the complexity of the decoding of CDE 40,1. The first algorithm is called the representation decoding algorithm. It is based on the pattern of codewords of E10. Using certain automorphisms of E10, we show that only eight types of codewords of E10 can produce all the codewords of E10. The second algorithm is called the syndrome decoding algorithm based on E10. It first solves the syndrome equation in E10 and finds a corresponding binary codeword of CDE 40,1.