  In this paper, we study the problem of constructing projective systematic
authentication schemes based on binary linear codes. In systematic
authentication, a tag for authentication is generated and then appended to the
information, also referred to as the source, to be sent from the sender.
Existing approaches to leverage projective constructions focus primarily on
codes over large alphabets, and the projection is simply into one single symbol
of the codeword. In this work, we extend the projective construction and
propose a general projection process in which the source, which is mapped to a
higher dimensional codeword in a given code, is first projected to a lower
dimensional vector. The resulting vector is then masked to generate the tag. To
showcase the new method, we focus on leveraging binary linear codes and, in
particular, Reed-Muller (RM) codes for the proposed projective construction.
More specifically, we propose systematic authentication schemes based on RM
codes, referred to as RM-Acodes. We provide analytical results for
probabilities of deception, widely considered as the main metrics to evaluate
the performance of authentication systems. Through our analysis, we discover
and discuss explicit connections between the probabilities of deception and
various properties of RM codes.
