Cyclic subcodes of generalized Reed-Muller codes

We define homogeneous generalised Reed-Muller codes (HRM codes). These are subcodes of the generalized Reed-Muller (GRM) codes. We show, that while they have less information symbols, there is a significant increase in the minimum distance. HRM codes are also naturally related to projective Reed-Muller codes (PRM codes). In particular, we can deduce their parameters from the PRM codes. We show, that punctured HRM codes are cyclic, while PRM codes in general are not. Hence, HRM codes compare favorably to both GRM codes and PRM codes. We show that binary trace codes of q-ary HRM codes are well-defined and give their parameters. The trace codes include codes that have the same length and minimum distance as some classical RM codes, but with a significant increase in the dimension