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.