Best Known (12, 12+38, s)-Nets in Base 4
(12, 12+38, 28)-Net over F4 — Constructive and digital
Digital (12, 50, 28)-net over F4, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 12 and N(F) ≥ 28, using
(12, 12+38, 29)-Net over F4 — Digital
Digital (12, 50, 29)-net over F4, using
- net from sequence [i] based on digital (12, 28)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 12 and N(F) ≥ 29, using
(12, 12+38, 56)-Net over F4 — Upper bound on s (digital)
There is no digital (12, 50, 57)-net over F4, because
- 2 times m-reduction [i] would yield digital (12, 48, 57)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(448, 57, F4, 36) (dual of [57, 9, 37]-code), but
- construction Y1 [i] would yield
- linear OA(447, 51, F4, 36) (dual of [51, 4, 37]-code), but
- OA(49, 57, S4, 6), but
- discarding factors would yield OA(49, 40, S4, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 273901 > 49 [i]
- discarding factors would yield OA(49, 40, S4, 6), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(448, 57, F4, 36) (dual of [57, 9, 37]-code), but
(12, 12+38, 58)-Net in Base 4 — Upper bound on s
There is no (12, 50, 59)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(450, 59, S4, 38), but
- the linear programming bound shows that M ≥ 12 890525 988420 026441 280523 449330 040832 / 8 204625 > 450 [i]