Characteristics of Hadamard square of special Reed — Muller subcodes

The existence of some structure in a code can lead to the decrease of security of the whole system built on it. Often subcodes are used to “disguise” the code as a “general-looking” one. However, the security of subcodes, whose Hadamard square is equal to the square of the original code, can be reduced to the security of this code. The paper finds the limiting conditions on the number of vectors of degree $ r $ whose removing retains this weakness for Reed — Muller subcodes and, accordingly, conditions for it to vanish. For $ r = 2 $ the exact structure of all resistant subcodes has been found. For an arbitrary code $\mathrm{RM}(r, m) $, the desired number of vectors to remove for providing the security has been estimated from both sides. Finally, the ratio of subcodes with Hadamard square unequal to the square of the original code has been proved to tend to zero if additional conditions on the codimension of the subcode and the parameter $ r $ are imposed and $ m \rightarrow \infty $. Thus, the implementation of checks proposed in the paper helps to immediately filter out some insecure subcodes.