weighted TOP ordering

[steenpass]

From Diego Ruano:

Dear Singular team,

[...]

For some list decoding algorithms (in coding theory), one uses the weighted TOP ordering for computing Gröbner bases over modules. Magma has implemented this term ordering (TOPW in Magma), but we would like to use Singular as usual.

​http://magma.maths.usyd.edu.au/magma/handbook/text/1147

We are wondering if you could implement this term ordering in Singular, please?

The formal definition of weighted TOP is:

Given weights w_1, ... w_q,
x^a e_i < x^b e_j
if and only if a + w_i < b + w_j
or if (a + w_i = b + w_j and i < j).

One can find the definition of this term ordering and an application in: ​http://arxiv.org/abs/1011.1040

Best regards,

Diego

[steenpass]

There is no predefined weighted TOP ordering in Singular. However, this can be emulated as follows:

Use extra variables y_1, ..., y_q and define the ordering by the matrix

1, w_1, ...,w_q
0,   1
0,     1
0,       ...
0,           1

If you want to make sure that the y_i only occur up to power 1, you can define a quotient ring by maxideal(2). Be careful not to allow any terms without y_i.