The weight distributions of cosets of the second-order Reed-Muller code of length 128 in the third-order Reed-Muller code of length 128

In this paper, cosets of the second order Reed-Muller code of length 2(m), denoted RM(m,2), in the third order Reed-Muller code of the same length, denoted RM(m,3), are studied. The set of cosets, RM(m,3)/RM(m,2) is partitioned into blocks. Two cosets are in the same block, if and only if there is a transformation in the general linear group by which one coset is transformed into the other. Two cosets in the same block have the same weight distribution. For the code length less than or equal to 128, the representative coset leader of each block is presented and the weight distribution of cosets in the block is computed. By using these results, the extended code of a cyclic code of length length 128 between RM(7,2) and RM(7,3) can be decomposed into a set of cosets in RM(7,3)/RM(7,2), and its weight distribution can be derived. Several cyclic codes of length 127 are shown to be equivalent and some new linear unequal error protection codes are found.