High-Dimensional Cross Parity Codes and Parities from Lower Than (d − 1)-Dimensional Hyperplanes

Cross parity codes are mostly used as 2-dimensional codes, and sometimes as 3-dimensional codes. We argue that higher dimensions can help to reduce the number of parity bits, and thus deserve further investigation. As a start, we investigate parities from (d−2)-dimensional hyperplanes in d-dimensional parity codes, instead of parities from (d−1)-dimensional hyperplanes as usual.