Best Known (43, 43+123, s)-Nets in Base 4
(43, 43+123, 56)-Net over F4 — Constructive and digital
Digital (43, 166, 56)-net over F4, using
- t-expansion [i] based on digital (33, 166, 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
(43, 43+123, 75)-Net over F4 — Digital
Digital (43, 166, 75)-net over F4, using
- t-expansion [i] based on digital (40, 166, 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
(43, 43+123, 223)-Net over F4 — Upper bound on s (digital)
There is no digital (43, 166, 224)-net over F4, because
- 3 times m-reduction [i] would yield digital (43, 163, 224)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4163, 224, F4, 120) (dual of [224, 61, 121]-code), but
- residual code [i] would yield OA(443, 103, S4, 30), but
- the linear programming bound shows that M ≥ 126192 248039 190038 466119 259112 020487 681472 723343 663318 526710 983775 014092 800000 / 1468 588736 152011 748631 330502 226638 946367 298107 382103 > 443 [i]
- residual code [i] would yield OA(443, 103, S4, 30), but
- extracting embedded orthogonal array [i] would yield linear OA(4163, 224, F4, 120) (dual of [224, 61, 121]-code), but
(43, 43+123, 286)-Net in Base 4 — Upper bound on s
There is no (43, 166, 287)-net in base 4, because
- 1 times m-reduction [i] would yield (43, 165, 287)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2504 917127 534056 675604 480367 928496 885614 624049 562855 624230 804549 150641 646773 723300 529858 439991 542640 > 4165 [i]