Best Known (186−142, 186, s)-Nets in Base 4
(186−142, 186, 56)-Net over F4 — Constructive and digital
Digital (44, 186, 56)-net over F4, using
- t-expansion [i] based on digital (33, 186, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(186−142, 186, 75)-Net over F4 — Digital
Digital (44, 186, 75)-net over F4, using
- t-expansion [i] based on digital (40, 186, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(186−142, 186, 185)-Net over F4 — Upper bound on s (digital)
There is no digital (44, 186, 186)-net over F4, because
- 6 times m-reduction [i] would yield digital (44, 180, 186)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4180, 186, F4, 136) (dual of [186, 6, 137]-code), but
- construction Y1 [i] would yield
- linear OA(4179, 183, F4, 136) (dual of [183, 4, 137]-code), but
- linear OA(46, 186, F4, 3) (dual of [186, 180, 4]-code or 186-cap in PG(5,4)), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4180, 186, F4, 136) (dual of [186, 6, 137]-code), but
(186−142, 186, 288)-Net in Base 4 — Upper bound on s
There is no (44, 186, 289)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 11363 448814 313307 050155 491940 047936 590631 399616 879617 429797 200523 193723 977969 395792 907161 230662 271486 681509 307520 > 4186 [i]