## Dual Code (with Bound on d by Construction Y1)

Shortening allows the construction of a linear [su, nu, d]-code from a linear [s, n, d]-code for u = 1,…, n. However, in some situations it is possible to obtain a code-dimension of nu + 1 instead of nu.

Given a linear [s, n, d]-code C, then a new linear [su, nu + 1, d]-code Cʹ over the same field exists, given that the dual distance d of C (the distance of C, the dual code of C) is less or equal to u.

The new code is obtained by removing u columns from the parity check matrix of C such that all d non-zero coordinates of the minimum weight code word of C are removed.

Since MinT has no knowledge about the minimum distance of C, it has to assume the worst and use the upper bound on the minimum distance of any [s, sn]-code.

### Parameters of the Involved Codes

This propagation rule is used by MinT in three different circumstances:

• The straightforward way is to obtain Cʹ based on C. This method is listed as non-constructive because there is no efficient method for determining the minimum-weight codeword in C. Cʹ is listed as parent of the construction.

• This method can also be used for showing that the minimum distance of C is greater than u, provided that Cʹ does not exist. Applied in this way this rule is constructive, because the dual code can always be constructed efficiently. MinT lists C is listed as parent, the non-existence of Cʹ as the second parent of this construction.

• When MinT uses this method for establishing bounds for codes, the non-existence is proven for the code C, whereas the first base code listed is the (non existent) resulting code Cʹ, and the second base code is the (non existent) dual C of C.