UGC NET CSE | January 2017 | Part 3 | Question: 65

Question

A t-error correcting q-nary linear code 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. $q^{-t}$

Answer 1

I think the answer is (B)

A linear code of length n and rank k is a linear subspace C with dimension k of the vector space {\displaystyle \mathbb {F} _{q}^{n}} where {\displaystyle \mathbb {F} _{q}} is the finite field with q elements. Such a code is called a q-ary code. If q = 2 or q = 3, the code is described as a binary code, or a ternary code respectively. The vectors in C are called codewords. The size of a code is the number of codewords and equals q^k

Reference:https://en.wikipedia.org/wiki/Linear_code

correct me if i am wrong

Comments

It seems like you are trying to describe a vector space over a finite field with q elements. The notation $\mathbb{F}_q^n$ denotes a vector space of dimension n over the field $\mathbb{F}_q$. This means that each element of the vector space is a list of n elements from the field $\mathbb{F}_q$. The field $\mathbb{F}_q$ is a set of q elements (usually taken to be the numbers 0 through q-1) that can be combined using the operations of addition, subtraction, and multiplication (just like the real numbers).

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Yes, that is correct. A linear code of length n and rank k is a subspace of the vector space $\mathbb{F}_q^n$ with dimension k. This means that it is a set of vectors (codewords) that can be combined using the vector space operations of addition and scalar multiplication, and that has k linearly independent basis vectors. The size of the code, or the number of codewords it contains, is equal to $q^k$.