Determinant sudoku

Some friends of mine have trouble finding sufficiently difficult sudoku puzzles to keep them occupied. So one day while daydreaming I came up with Determinant Sudoku, in which you take the determinant of each square within a valid sudoku and make a super-square in which no row or column repeats a determinant.

In base-4 sudoku, such a Determinant Sudoku is a grid of 12x12 numbers, not too much larger than a base-10 ordinary sudoku. But I wasn't sure if any valid base-4 Determinant Sudoku actually existed, so I wrote a program to help me check.

I did manage to construct a valid base-4 Determinant Sudoku, but I'm not sure how many others exist.

Here is the source code to enumerate all base-4 sudoku and print their determinants, and I've copied the comment header below which includes more explanation and the base-4 determinant sudoku I found:

// This program enumerates all the base-4 sudoku and calculates
// their determinants.
//
// The idea is this: start with base-4 sudoku, so that you get
// 2x2 squares which have the numbers 0,1,2,3 with none repeated.
// Create a 2x2 grid of those 2x2 squares, so that each row and column
// has the numbers 0,1,2,3.  (Ordinary sudoku, but 4x4 instead of 9x9).
//
// Here's an example of a valid base-4 sudoku:
// 01 23 
// 23 01 
//
// 10 32 
// 32 10 
//
// There appear to be 288 distint base-4 sudoku, without removing isomorphisms.
//
// My idea was to take the determinant of each of those 2x2 matrices and create
// a "determinant sudoku".
//
// In matrix math, the determinant of
// a b
// c d
// is a*d - b*c.
//
// There are six possible determinants of the 2x2 blocks.
//
// To construct a base-4 determinant-sudoku, make a 3x3 grid of 4x4 base-4
// sudoku blocks.  There will be six rows and six columns of 2x2 squares.
//
// How many base-4 determinant-sudoku blocks exist? 
// This program almost answers that question, but not quite.
//
// Of the 288 base-4 sudoku, 168 have the same determinant, leaving 120
// base-4 sudoku that could contribute to a base-4 determinant-sudoku.
//
// I constructed one base-4 determinant-sudoku by hand:
//
// 01 23  |  01 32  | 02 31
// 23 01  |  32 01  | 31 02
//        |         |
// 10 32  |  10 23  | 20 13
// 32 10  |  23 10  | 13 20
// 
// ----------------------
// 
// 01 32  |  02 31  | 01 23   
// 32 01  |  31 02  | 23 01   
//        |         |
// 10 23  |  20 13  | 10 32   
// 23 10  |  13 20  | 32 10   
// 
// ----------------------
// 
// 02 31  |  01 23  | 01 32  
// 31 02  |  23 01  | 32 01  
//        |         |  
// 20 13  |  10 32  | 10 23  
// 13 20  |  32 10  | 23 10  
// 
// 
// Determinants:
// 
// -2  2   -3  3   -6  6
//  2 -2    3 -3    6 -6
// 
// -3  3   -6  6   -2  2
//  3 -3    6 -6    2 -2
// 
// -6  6   -2  2   -3  3
//  6 -6    2 -2    3 -3
// 
// Each 4x4 block is a valid base-4 sudoku.  And the 6x6 grid of 2x2 blocks
// has no repeated determinants in any row or column.
//
// (This base-4 determinant sudoku isn't very nice, since it repeats the
// same 3 base-4 sudoku 3 times each.)
//
// So the question remains open: how many base-4 determinant sudoku are there?