Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 3K / i

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- Solution i

Solution

For a binary block code,

n

is the common length of its codewords,

m = ∣ C ∣

is the number of codewords, and

d = min_{x, y \in C x \neq = y} d_{H} (x, y)

is its minimum Hamming distance.

The parity extension is

C^{+} = {(x_{1}, \dots, x_{n}, \sum_{j = 1}^{n} x_{j}) : x \in C} .

Every extended word has even Hamming weight. The distance between two words increases by one exactly when their original distance is odd, so

C^{+} has parameters [n + 1, m, d^{+}], d^{+} = {d, d + 1, d even, d odd .

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