Reed-Muller codes, elementary symmetric functions and asymmetric error correction

This paper shows that the first order Reed-Muller codes punctured in one component fall into a class of t-asymmetric error correcting (t-AEC) codes with very fast decoding. Hence, these linear Reed-Muller codes give a nice example of t-AEC codes which are very simple to both encode and decode. Decoding of these codes is much simpler than the usual t-SEC BCH code decoding because the syndromes, which are based on elementary symmetric functions of the received word, directly give the number of errors and the error locator polynomial. The result is based on some interesting properties which are proven in general for geometry codes.