Question

2. Given linear code $C \subseteq GF(q)^n$ with $k$ dimension. (a) Define dual code $C^\perp$ dan what is dimension from $C^\perp$? (b) If $q = 2$, that $C \subseteq GF(2)^n = \mathbb{Z}_2^n$, and known that binary linear code $C$ self dual, that is $C^\perp = C$: i. Show That $n$ must be an even number. ii. Show that every codeword from $C$ have even weight. iii. Show that vector 111...1 with weight $n$ are in $C$.

          2. Given linear code $C \subseteq GF(q)^n$ with $k$ dimension.
(a) Define dual code $C^\perp$ dan what is dimension from $C^\perp$?
(b) If $q = 2$, that $C \subseteq GF(2)^n = \mathbb{Z}_2^n$, and known that binary linear code $C$ self dual, that is $C^\perp = C$:
i. Show That $n$ must be an even number.
ii. Show that every codeword from $C$ have even weight.
iii. Show that vector 111...1 with weight $n$ are in $C$.

2. Given linear code C ⊆ GF(q)^n with k dimension.
(a) Define dual code C^⊥ dan what is dimension from C^⊥?
(b) If q = 2, that C ⊆ GF(2)^n = ℤ2^n, and known that binary linear code C self dual, that is C^⊥ = C:
i. Show That n must be an even number.
ii. Show that every codeword from C have even weight.
iii. Show that vector 111...1 with weight n are in C.

Added by Anna W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

05/08/2024

Step 1

Step 1: The dual code $C^\perp$ of a linear code $C$ is the set of all vectors in $GF(q)^n$ that are orthogonal to every vector in $C$. Show more…

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2. Given linear code $C \subseteq GF(q)^n$ with $k$ dimension. (a) Define dual code $C^\perp$ dan what is dimension from $C^\perp$? (b) If $q = 2$, that $C \subseteq GF(2)^n = \mathbb{Z}_2^n$, and known that binary linear code $C$ self dual, that is $C^\perp = C$: i. Show That $n$ must be an even number. ii. Show that every codeword from $C$ have even weight. iii. Show that vector $111...1$ with weight $n$ are in $C$.