New classes of affine-invariant codes sandwiched between Reed–Muller codes

Reed–Muller codes have been extensively studied due to their favorable theoretical and mathematical properties. We propose some new classes of extended cyclic codes sandwiched between generalized Reed–Muller codes Rq(r−1,n) and Rq(r,n) with n=3m, m⩾1. We show that these new codes are affine-invariant, and compute their dimensions and dual codes. We also give lower bounds for their minimum distances. In addition, we explicitly determine the minimum vectors of these codes for some special cases. Our new method can be used to construct codes of length qn with odd n, which makes up for the shortcomings of the previous construction method which works only for even n. Moreover, our new extended cyclic codes have higher rate than those in Xu et al. (2023) [17].