Best Known (197−142, 197, s)-Nets in Base 4
(197−142, 197, 66)-Net over F4 — Constructive and digital
Digital (55, 197, 66)-net over F4, using
- t-expansion [i] based on digital (49, 197, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second 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) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(197−142, 197, 91)-Net over F4 — Digital
Digital (55, 197, 91)-net over F4, using
- t-expansion [i] based on digital (50, 197, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(197−142, 197, 326)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 197, 327)-net over F4, because
- 2 times m-reduction [i] would yield digital (55, 195, 327)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4195, 327, F4, 140) (dual of [327, 132, 141]-code), but
- residual code [i] would yield OA(455, 186, S4, 35), but
- the linear programming bound shows that M ≥ 2227 845699 810146 058122 102603 602007 534656 013828 852609 458125 647380 480000 / 1 700436 329479 757404 993871 394563 863537 > 455 [i]
- residual code [i] would yield OA(455, 186, S4, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(4195, 327, F4, 140) (dual of [327, 132, 141]-code), but
(197−142, 197, 369)-Net in Base 4 — Upper bound on s
There is no (55, 197, 370)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 41547 235088 435879 246952 449104 010012 301225 940288 480864 162803 131985 663424 824946 459069 236334 121329 557382 852610 985919 332904 > 4197 [i]