Best Known (70, 70+186, s)-Nets in Base 4
(70, 70+186, 66)-Net over F4 — Constructive and digital
Digital (70, 256, 66)-net over F4, using
- t-expansion [i] based on digital (49, 256, 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
(70, 70+186, 105)-Net over F4 — Digital
Digital (70, 256, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
(70, 70+186, 375)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 256, 376)-net over F4, because
- 2 times m-reduction [i] would yield digital (70, 254, 376)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 376, F4, 184) (dual of [376, 122, 185]-code), but
- residual code [i] would yield OA(470, 191, S4, 46), but
- the linear programming bound shows that M ≥ 30669 687295 767807 163249 547418 721993 874758 654208 012151 612633 672044 408069 277192 616104 296206 434138 576650 240000 / 20720 048409 770029 043765 568578 931955 503324 140138 849174 037831 490849 > 470 [i]
- residual code [i] would yield OA(470, 191, S4, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 376, F4, 184) (dual of [376, 122, 185]-code), but
(70, 70+186, 463)-Net in Base 4 — Upper bound on s
There is no (70, 256, 464)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 15952 125605 176411 676147 402035 226339 487515 599224 224628 984871 896194 723698 155761 232805 270154 523112 750919 562220 684090 008526 866754 479138 487019 981405 193326 828920 > 4256 [i]