Best Known (233−171, 233, s)-Nets in Base 4
(233−171, 233, 66)-Net over F4 — Constructive and digital
Digital (62, 233, 66)-net over F4, using
- t-expansion [i] based on digital (49, 233, 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
(233−171, 233, 99)-Net over F4 — Digital
Digital (62, 233, 99)-net over F4, using
- t-expansion [i] based on digital (61, 233, 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
(233−171, 233, 318)-Net over F4 — Upper bound on s (digital)
There is no digital (62, 233, 319)-net over F4, because
- 3 times m-reduction [i] would yield digital (62, 230, 319)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4230, 319, F4, 168) (dual of [319, 89, 169]-code), but
- residual code [i] would yield OA(462, 150, S4, 42), but
- the linear programming bound shows that M ≥ 1 659762 510567 225532 778073 956682 851160 927584 506340 544111 988906 878425 443399 317508 383360 785706 508045 270473 316579 147776 / 70196 277944 430470 237066 876146 512897 904291 630830 313677 137903 195390 483423 677609 > 462 [i]
- residual code [i] would yield OA(462, 150, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4230, 319, F4, 168) (dual of [319, 89, 169]-code), but
(233−171, 233, 409)-Net in Base 4 — Upper bound on s
There is no (62, 233, 410)-net in base 4, because
- 1 times m-reduction [i] would yield (62, 232, 410)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 56 610683 071390 878694 105468 100557 978983 351322 367850 504095 643351 031364 264578 472545 244559 438286 935774 795357 472030 569551 867652 955268 553658 421020 > 4232 [i]