Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 11G / c

Table of contents
- Solution c

Solution

Apply part (b) repeatedly. After the first puncturing, the code has rank

k - 1

n - d

, and minimum distance at least

⌈ d /2 ⌉

. Repeating with a minimum-weight codeword at each stage gives

n \geq d + ⌈ \frac{d}{2} ⌉ + ⌈ \frac{d}{2 ^{2}} ⌉ + c d o t s + ⌈ \frac{d}{2 ^{k - 1}} ⌉,

because nested ceilings obey

⌈ \frac{⌈ d / 2 ^{j} ⌉}{2} ⌉ = ⌈ \frac{d}{2 ^{j + 1}} ⌉ .

Thus

n \geq d + 1 \leq l \leq k - 1 \sum ⌈ \frac{d}{2 ^{l}} ⌉,

which is the binary Griesmer bound.

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