## Construction X with Algebraic-Geometric NRT-Codes

Construction X for NRT-codes allows the construction of a new NRT-code based on two linear NRT-codes C1C2, such that the dimension of C2 and the minimum distance of C1 is obtained. This is bought by increasing the length of the resulting code by the length of an additional code Ce, which has to be chosen depending on the parameters of C1 and C2.

Given a linear [(s, T ), n1, d1]-code C1, which is a subcode of a linear [(s, T ), n2, d2]-code C2, as well as an additional linear [(se, T ), ne, de]-code Ce, all over the same field, a new linear [(s + se, T ), n1 + ne, d2 + de]-code can be constructed provided that n1 + nen2 and d2 + ded1. Usually Ce will be chosen such that n2 = n1 + ne and d1 = d2 + de, however in some situations a smaller value of ne or de may also yield good results.

### Construction

Let G1, G2, and Ge denote the generator matrices of C1, C2, and Ce, respectively, such that the rows of G1 are a subset of the rows of G2. Let G2ʹ denote the ne×s matrix consisting of ne rows of G2 that are not in G1. Then the new code is defined by the (n2 + ne)×((s + se), T ) generator matrix   .

All code words formed by a non-trivial linear combination of the first n1 vectors have a weight of at least d1 because these are essentially the code words of C1 with seT additional 0’s appended. All other non-zero code words have a weight of at least d2 + de because they are built using a non-zero code word from C2 next to a non-zero code word from Ce.

### Applications

MinT applies construction X in the following situations: