Best Known (67, 67+174, s)-Nets in Base 4
(67, 67+174, 66)-Net over F4 — Constructive and digital
Digital (67, 241, 66)-net over F4, using
- t-expansion [i] based on digital (49, 241, 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+174, 99)-Net over F4 — Digital
Digital (67, 241, 99)-net over F4, using
- t-expansion [i] based on digital (61, 241, 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+174, 379)-Net over F4 — Upper bound on s (digital)
There is no digital (67, 241, 380)-net over F4, because
- 2 times m-reduction [i] would yield digital (67, 239, 380)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 380, F4, 172) (dual of [380, 141, 173]-code), but
- residual code [i] would yield OA(467, 207, S4, 43), but
- the linear programming bound shows that M ≥ 28014 561764 993307 241664 052911 871857 702310 880709 902206 525389 902012 241710 106857 807269 754280 270102 528000 / 1 234731 917948 789567 037359 176277 481011 072922 589066 284983 746001 > 467 [i]
- residual code [i] would yield OA(467, 207, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 380, F4, 172) (dual of [380, 141, 173]-code), but
(67, 67+174, 446)-Net in Base 4 — Upper bound on s
There is no (67, 241, 447)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 13 931476 613696 215026 809292 416429 770365 386514 501835 739244 062899 891580 769084 222349 038826 667149 271606 624050 078297 253375 889553 159767 971102 205857 823400 > 4241 [i]