Best Known (67, 67+179, s)-Nets in Base 4
(67, 67+179, 66)-Net over F4 — Constructive and digital
Digital (67, 246, 66)-net over F4, using
- t-expansion [i] based on digital (49, 246, 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+179, 99)-Net over F4 — Digital
Digital (67, 246, 99)-net over F4, using
- t-expansion [i] based on digital (61, 246, 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+179, 361)-Net over F4 — Upper bound on s (digital)
There is no digital (67, 246, 362)-net over F4, because
- 3 times m-reduction [i] would yield digital (67, 243, 362)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4243, 362, F4, 176) (dual of [362, 119, 177]-code), but
- residual code [i] would yield OA(467, 185, S4, 44), but
- the linear programming bound shows that M ≥ 27 922736 614479 963563 498030 039555 280180 497392 114764 937434 686895 163379 799248 329552 391127 539087 989670 936576 / 1212 198916 755108 948361 633672 026124 789528 916769 954260 490443 945723 > 467 [i]
- residual code [i] would yield OA(467, 185, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4243, 362, F4, 176) (dual of [362, 119, 177]-code), but
(67, 67+179, 443)-Net in Base 4 — Upper bound on s
There is no (67, 246, 444)-net in base 4, because
- 1 times m-reduction [i] would yield (67, 245, 444)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3278 424737 262322 591866 494083 774736 120825 367484 644165 883183 151953 920163 538523 839121 750028 742585 100398 393510 588288 009249 834724 563574 489864 824629 631530 > 4245 [i]