Best Known (231−165, 231, s)-Nets in Base 4
(231−165, 231, 66)-Net over F4 — Constructive and digital
Digital (66, 231, 66)-net over F4, using
- t-expansion [i] based on digital (49, 231, 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
(231−165, 231, 99)-Net over F4 — Digital
Digital (66, 231, 99)-net over F4, using
- t-expansion [i] based on digital (61, 231, 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
(231−165, 231, 403)-Net over F4 — Upper bound on s (digital)
There is no digital (66, 231, 404)-net over F4, because
- 1 times m-reduction [i] would yield digital (66, 230, 404)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4230, 404, F4, 164) (dual of [404, 174, 165]-code), but
- residual code [i] would yield OA(466, 239, S4, 41), but
- the linear programming bound shows that M ≥ 222 701100 724979 298348 829615 805692 012896 998205 225198 824371 851659 244379 751219 018386 636800 / 38358 477730 413015 488846 745362 992009 108847 310917 > 466 [i]
- residual code [i] would yield OA(466, 239, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4230, 404, F4, 164) (dual of [404, 174, 165]-code), but
(231−165, 231, 445)-Net in Base 4 — Upper bound on s
There is no (66, 231, 446)-net in base 4, because
- 1 times m-reduction [i] would yield (66, 230, 446)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3 241620 881798 888221 700598 728457 836295 103587 326822 119860 421455 208712 245928 190857 166618 729444 775541 446290 488732 855861 280040 565953 136406 049060 > 4230 [i]