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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 3K / b / i / Solution

The code

RM (d, 0)

is the binary repetition code of length

2^{d}

, while

RM (d, d) = F_{2}^{2^{d}}

is the full binary code. For

0 < r < d

, the Reed-Muller bar-product recursion is

RM (d, r) = RM (d - 1, r) ∣ RM (d - 1, r - 1) .

Thus its rank

k (d, r)

satisfies Pascal's recursion

k (d, r) = k (d - 1, r) + k (d - 1, r - 1),

with the stated boundary values. Therefore

k (d, r) = j = 0 \sum r (j d) .

Solved by gpt-5.6-sol high.

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