Best Known (57, 202, s)-Nets in Base 4
(57, 202, 66)-Net over F4 — Constructive and digital
Digital (57, 202, 66)-net over F4, using
- t-expansion [i] based on digital (49, 202, 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
(57, 202, 91)-Net over F4 — Digital
Digital (57, 202, 91)-net over F4, using
- t-expansion [i] based on digital (50, 202, 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
(57, 202, 339)-Net over F4 — Upper bound on s (digital)
There is no digital (57, 202, 340)-net over F4, because
- 1 times m-reduction [i] would yield digital (57, 201, 340)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4201, 340, F4, 144) (dual of [340, 139, 145]-code), but
- residual code [i] would yield OA(457, 195, S4, 36), but
- the linear programming bound shows that M ≥ 494 490067 436667 455875 598203 818411 591542 653847 684514 372082 636620 702806 966272 / 22309 972171 916724 959618 830756 395146 773299 > 457 [i]
- residual code [i] would yield OA(457, 195, S4, 36), but
- extracting embedded orthogonal array [i] would yield linear OA(4201, 340, F4, 144) (dual of [340, 139, 145]-code), but
(57, 202, 385)-Net in Base 4 — Upper bound on s
There is no (57, 202, 386)-net in base 4, because
- 1 times m-reduction [i] would yield (57, 201, 386)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 12 072154 740893 860680 503668 635395 582364 270436 662646 636628 962624 963729 682252 092437 229984 920359 043426 134720 936900 287776 546736 > 4201 [i]