Best Known (73, 73+177, s)-Nets in Base 4
(73, 73+177, 104)-Net over F4 — Constructive and digital
Digital (73, 250, 104)-net over F4, using
- net from sequence [i] based on digital (73, 103)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 104, using
- F6 from the 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) = 73 and N(F) ≥ 104, using
(73, 73+177, 112)-Net over F4 — Digital
Digital (73, 250, 112)-net over F4, using
- net from sequence [i] based on digital (73, 111)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 112, using
(73, 73+177, 465)-Net over F4 — Upper bound on s (digital)
There is no digital (73, 250, 466)-net over F4, because
- 1 times m-reduction [i] would yield digital (73, 249, 466)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4249, 466, F4, 176) (dual of [466, 217, 177]-code), but
- residual code [i] would yield OA(473, 289, S4, 44), but
- the linear programming bound shows that M ≥ 6795 366876 081531 133774 121792 921977 453856 545502 548417 574102 431690 872138 283810 785820 606988 288000 / 74 611970 849216 063476 829413 418651 211240 390198 261413 > 473 [i]
- residual code [i] would yield OA(473, 289, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4249, 466, F4, 176) (dual of [466, 217, 177]-code), but
(73, 73+177, 495)-Net in Base 4 — Upper bound on s
There is no (73, 250, 496)-net in base 4, because
- 1 times m-reduction [i] would yield (73, 249, 496)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 853340 999748 049444 723092 534395 951498 531968 711111 366176 162465 866738 782028 141294 073103 568434 742594 282635 545650 979379 126221 232886 118400 432936 689974 872054 > 4249 [i]