Best Known (67, 67+166, s)-Nets in Base 4
(67, 67+166, 66)-Net over F4 — Constructive and digital
Digital (67, 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
(67, 67+166, 99)-Net over F4 — Digital
Digital (67, 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
(67, 67+166, 421)-Net over F4 — Upper bound on s (digital)
There is no digital (67, 233, 422)-net over F4, because
- 2 times m-reduction [i] would yield digital (67, 231, 422)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4231, 422, F4, 164) (dual of [422, 191, 165]-code), but
- residual code [i] would yield OA(467, 257, S4, 41), but
- the linear programming bound shows that M ≥ 188282 701092 137183 285392 695738 133645 870848 581784 362420 528688 778251 423906 057085 255680 / 8 301637 427320 826521 830664 836391 530003 316903 > 467 [i]
- residual code [i] would yield OA(467, 257, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4231, 422, F4, 164) (dual of [422, 191, 165]-code), but
(67, 67+166, 452)-Net in Base 4 — Upper bound on s
There is no (67, 233, 453)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 210 660096 912400 317766 309503 977141 160529 082697 023017 431772 426616 760565 200493 774740 242496 761442 468093 539645 783846 879805 324194 052486 607560 759520 > 4233 [i]