Best Known (215−155, 215, s)-Nets in Base 4
(215−155, 215, 66)-Net over F4 — Constructive and digital
Digital (60, 215, 66)-net over F4, using
- t-expansion [i] based on digital (49, 215, 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
(215−155, 215, 91)-Net over F4 — Digital
Digital (60, 215, 91)-net over F4, using
- t-expansion [i] based on digital (50, 215, 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
(215−155, 215, 353)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 215, 354)-net over F4, because
- 3 times m-reduction [i] would yield digital (60, 212, 354)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4212, 354, F4, 152) (dual of [354, 142, 153]-code), but
- residual code [i] would yield OA(460, 201, S4, 38), but
- the linear programming bound shows that M ≥ 73089 680874 254803 480484 380402 731584 606836 512208 238432 667847 363593 365796 276173 537280 / 52941 428031 033877 887534 646852 798421 725736 559559 > 460 [i]
- residual code [i] would yield OA(460, 201, S4, 38), but
- extracting embedded orthogonal array [i] would yield linear OA(4212, 354, F4, 152) (dual of [354, 142, 153]-code), but
(215−155, 215, 402)-Net in Base 4 — Upper bound on s
There is no (60, 215, 403)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 214, 403)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 734 456587 188029 868799 428791 935988 588027 851998 121190 002808 499160 506496 925829 594391 245035 381303 005310 449247 080347 515957 257007 473264 > 4214 [i]