Another generalisation of the binary Reed-Muller codes and its applications

The punctured binary Reed-Muller code is cyclic and was generalised into the punctured generalised Reed-Muller code over GF(q) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed-Muller code and the binary Reed-Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [3m−1,3m−1−2m,4] for all m≥2. Their extended codes have parameters [3m,3m−1−2m,5] for all m≥2, and are also optimal.