what is the intersection of two linear codes?

I have a question about linear codes. Let's say we have two (n,k) linear codes C1 and C2 with parity check matrix H1 and H2. Is the intersection of C1 and C2 still a linear code? If so, what is its parity check matrix H3 given H1 and H2? C3 is the intersection of C1 and C2 means H1c3=0 and H2c3=0 for all c3\in C3.

Answers


Yes. It is also a linear code.

A linear code of length n and rank k is a linear subspace C with dimension k of the vector space V.

Given subspaces U and W of a vector space V, then their intersection UW := {v ∈ V : v is an element of both U and W} is also a subspace of V.

To obtain H dimension this statement may be used:

Let (G,+G,∘)K be a K-vector space. Let M and N be finite-dimensional subspaces of G.

Then M+N and M∩N are finite-dimensional, and:

dim(M+N) + dim(M∩N) = dim(M) + dim(N)

so:

dim(M+N) + dim(M∩N) = k1 + k2

where dim(M∩N) is new k of the intersection.


Need Your Help

What is @interface UITableView (MyTableViewDelegate)

iphone objective-c cocoa ipad

Im looking at some source code written by someone else and its intrigues me to see this line:

Bootstrap Panel Styling

html css twitter-bootstrap

I am using Legacy Bootstrap 2.3 and I am having some trouble getting some CSS to work correctly.

About UNIX Resources Network

Original, collect and organize Developers related documents, information and materials, contains jQuery, Html, CSS, MySQL, .NET, ASP.NET, SQL, objective-c, iPhone, Ruby on Rails, C, SQL Server, Ruby, Arrays, Regex, ASP.NET MVC, WPF, XML, Ajax, DataBase, and so on.