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mathlinear-algebraerror-detection

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.

Solution

  • 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.