Best Known (250−181, 250, s)-Nets in Base 4
(250−181, 250, 66)-Net over F4 — Constructive and digital
Digital (69, 250, 66)-net over F4, using
- t-expansion [i] based on digital (49, 250, 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
(250−181, 250, 99)-Net over F4 — Digital
Digital (69, 250, 99)-net over F4, using
- t-expansion [i] based on digital (61, 250, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(250−181, 250, 375)-Net over F4 — Upper bound on s (digital)
There is no digital (69, 250, 376)-net over F4, because
- 1 times m-reduction [i] would yield digital (69, 249, 376)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4249, 376, F4, 180) (dual of [376, 127, 181]-code), but
- residual code [i] would yield OA(469, 195, S4, 45), but
- the linear programming bound shows that M ≥ 82 828010 425088 398135 365726 677314 464149 340113 774380 345347 664035 522400 039552 947596 743692 557981 180235 874304 000000 / 227 578582 237535 291818 752893 814619 068871 044126 923591 235904 712727 792773 > 469 [i]
- residual code [i] would yield OA(469, 195, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4249, 376, F4, 180) (dual of [376, 127, 181]-code), but
(250−181, 250, 458)-Net in Base 4 — Upper bound on s
There is no (69, 250, 459)-net in base 4, because
- 1 times m-reduction [i] would yield (69, 249, 459)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 865613 934990 415674 642318 278576 717069 579509 633708 442532 909296 922724 312106 900526 411546 906319 282185 073487 169338 144531 217513 397899 507637 415089 480313 327656 > 4249 [i]