Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 11H / b / iv

Table of contents
- Solution iv

Solution

Puncturing gives a binary code of length

23

2^{12}

, and distance at least

7

. Radius-three balls are disjoint and

2^{12} \sum_{j = 0}^{3} (j 23) = 2^{12} (1 + 23 + 253 + 1771) = 2^{23} .

The Hamming bound is an equality, so

G^{-}

is perfect.

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