A t-error correcting q-nary linear code must satisfy :…
2017
A t-error correcting q-nary linear code must satisfy :
\(M\sum_{i=0}^{t}(\frac{n}{i})(q-1)^{i}\leq X\)
Where M is the number of code words and X is
- A.
\(q^n\) - B.
\(q^t\) - C.
\(q^{-n}\) - D.
\(q^{-t}\)
Attempted by 20 students.
Show answer & explanation
Correct answer: A
Answer: X = q^n
Explanation:
Total number of q-ary vectors of length n (the whole vector space) is q^n.
A Hamming ball of radius t around any codeword contains sum_{i=0}^t (n choose i) * (q-1)^i vectors.
In a t-error-correcting code, Hamming balls of radius t centered at distinct codewords are disjoint.
Therefore, the number of codewords M multiplied by the ball size cannot exceed the whole space: M * sum_{i=0}^t (n choose i) * (q-1)^i ≤ q^n.
Hence X is q^n.
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