Best Known (69, 69+26, s)-Nets in Base 4
(69, 69+26, 384)-Net over F4 — Constructive and digital
Digital (69, 95, 384)-net over F4, using
- 1 times m-reduction [i] based on digital (69, 96, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 32, 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, 32, 128)-net over F64, using
(69, 69+26, 387)-Net in Base 4 — Constructive
(69, 95, 387)-net in base 4, using
- 1 times m-reduction [i] based on (69, 96, 387)-net in base 4, using
- trace code for nets [i] based on (5, 32, 129)-net in base 64, using
- 3 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- base change [i] based on digital (0, 30, 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, 30, 129)-net over F128, using
- 3 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- trace code for nets [i] based on (5, 32, 129)-net in base 64, using
(69, 69+26, 671)-Net over F4 — Digital
Digital (69, 95, 671)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(495, 671, F4, 26) (dual of [671, 576, 27]-code), using
- 575 step Varšamov–Edel lengthening with (ri) = (7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 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, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0) [i] based on linear OA(426, 27, F4, 26) (dual of [27, 1, 27]-code or 27-arc in PG(25,4)), using
- dual of repetition code with length 27 [i]
- 575 step Varšamov–Edel lengthening with (ri) = (7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 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, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0) [i] based on linear OA(426, 27, F4, 26) (dual of [27, 1, 27]-code or 27-arc in PG(25,4)), using
(69, 69+26, 47408)-Net in Base 4 — Upper bound on s
There is no (69, 95, 47409)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1569 338617 475939 298524 979265 711452 116205 234822 416158 492480 > 495 [i]