Best Known (1, 1+5, s)-Nets in Base 4
(1, 1+5, 9)-Net over F4 — Constructive and digital
Digital (1, 6, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
(1, 1+5, 11)-Net over F4 — Upper bound on s (digital)
There is no digital (1, 6, 12)-net over F4, because
- 1 times m-reduction [i] would yield digital (1, 5, 12)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(45, 12, F4, 4) (dual of [12, 7, 5]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(47, 12, F4, 6) (dual of [12, 5, 7]-code), but
- discarding factors / shortening the dual code would yield linear OA(47, 10, F4, 6) (dual of [10, 3, 7]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(47, 12, F4, 6) (dual of [12, 5, 7]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(45, 12, F4, 4) (dual of [12, 7, 5]-code), but
(1, 1+5, 12)-Net in Base 4 — Upper bound on s
There is no (1, 6, 13)-net in base 4, because
- extracting embedded OOA [i] would yield OOA(46, 13, S4, 2, 5), but
- the linear programming bound for OOAs shows that M ≥ 225280 / 53 > 46 [i]