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Let e: B˄m→B˄n be a group code. The minimum distance of ‘e’ is equal to:
(A) the maximum weight of a non zero code word
(B) the minimum weight of a non zero code word
(C) m
(D) n
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2 Answers

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Option B

The minimum weight of a non zero code word
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Linear codes
The binary alphabet B = {0, 1} is naturally identified with
Z2, the field of 2 elements. Then Bn
can be regarded as the
n-dimensional vector space over the field Z2.
A binary code f : Bm → Bn
is linear if f is a linear
transformation of vector spaces. Any linear code is given by a
generator matrix G, which is an m×n matrix with entries
from Z2 such that f (w) = wG (here w is regarded as a row
vector). For a systematic code, G is of the form (Im|A).
Theorem If f : Bm → Bn
is a linear code, then
• the set W of all codewords forms a subspace (and a
subgroup) of Bn
;
• the zero word is a codeword;
• the minimum distance between distinct codewords is equal
to the minimum weight of nonzero codewords.