Best Known (84, 84+33, s)-Nets in Base 4
(84, 84+33, 384)-Net over F4 — Constructive and digital
Digital (84, 117, 384)-net over F4, using
- t-expansion [i] based on digital (83, 117, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 39, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 39, 128)-net over F64, using
(84, 84+33, 387)-Net in Base 4 — Constructive
(84, 117, 387)-net in base 4, using
- trace code for nets [i] based on (6, 39, 129)-net in base 64, using
- 3 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
- base change [i] based on digital (0, 36, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 36, 129)-net over F128, using
- 3 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
(84, 84+33, 694)-Net over F4 — Digital
Digital (84, 117, 694)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4117, 694, F4, 33) (dual of [694, 577, 34]-code), using
- 576 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0) [i] based on linear OA(433, 34, F4, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,4)), using
- dual of repetition code with length 34 [i]
- 576 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0) [i] based on linear OA(433, 34, F4, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,4)), using
(84, 84+33, 52510)-Net in Base 4 — Upper bound on s
There is no (84, 117, 52511)-net in base 4, because
- 1 times m-reduction [i] would yield (84, 116, 52511)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6902 762862 167680 610529 882615 066606 485295 850818 110459 276496 600396 732835 > 4116 [i]