Best Known (42, 42+106, s)-Nets in Base 4
(42, 42+106, 56)-Net over F4 — Constructive and digital
Digital (42, 148, 56)-net over F4, using
- t-expansion [i] based on digital (33, 148, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 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) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(42, 42+106, 75)-Net over F4 — Digital
Digital (42, 148, 75)-net over F4, using
- t-expansion [i] based on digital (40, 148, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(42, 42+106, 276)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 148, 277)-net over F4, because
- 2 times m-reduction [i] would yield digital (42, 146, 277)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4146, 277, F4, 104) (dual of [277, 131, 105]-code), but
- residual code [i] would yield OA(442, 172, S4, 26), but
- the linear programming bound shows that M ≥ 34 487674 602431 075505 978819 529968 455340 679044 792320 / 1 686703 652728 844301 656717 > 442 [i]
- residual code [i] would yield OA(442, 172, S4, 26), but
- extracting embedded orthogonal array [i] would yield linear OA(4146, 277, F4, 104) (dual of [277, 131, 105]-code), but
(42, 42+106, 287)-Net in Base 4 — Upper bound on s
There is no (42, 148, 288)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 128680 410551 731603 292114 670156 737881 407326 842181 248968 250962 136084 863168 532450 007171 423800 > 4148 [i]