Code Products and Bilinear Maps Sample Clauses

Code Products and Bilinear Maps. ‌ In this section we introduce a new perspective on codes which will be partic- ularly useful to analyze the parameters of code products. ⊆ '→ | · | Let C Fn be a code of dimension k, and let G be a generator matrix of C. As anticipated in Section 2.5, G defines in a natural way a linear embedding Fk → Fn whose image is C, namely the map x '→ xG. If we denote the columns of G by y1, . . . , yn, then the entries of the codeword xG are the inner products (x yi), for i = 1, . . . , n. Recall that, by classical linear algebra, the map x (x ) gives an isomorphism between any vector space and its dual. It follows that the image of the linear map (Fk)∗ Fn φ (φ(y1), . . . , φ(yn)) equals C and in particular is independent of the choice of the yi’s. Conversely, given any F-vector space V and v1, . . . , vn ∈ V , the image of the linear map V ∗ Fn φ (φ(v1), . . . , φ(vn)) is a code and, provided that the yi’s generate V , has dimension dim V . Let now C, D ⊆ Fn be two codes of dimension kC and kD respectively, and let y1, . . . , yn ∈ FkC and z1, . . . , zn ∈ FkD be the columns of a generator matrix of C and of a generator matrix of D respectively. As above, we can view the codes C and D as images of evaluation maps. In addition, we define the map Bil(FkC × FkD ) Fn B (B(y1, z1), . . . , B(yn, zn)) where Bil(FkC × FkD ) denotes the F-vector space of all bilinear forms FkC × FkD → F. This map is linear as well and its image is independent of the choice of yi’s and zi’s and equals the code product CD. In the symmetric case C = D we obtain the code square C2 as the image of the evaluation map evC : Quad(FkC ) Fn.