# 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 **U** ∩ **W** := {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.