 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/3k/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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)

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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