Best Known (211−55, 211, s)-Nets in Base 4
(211−55, 211, 531)-Net over F4 — Constructive and digital
Digital (156, 211, 531)-net over F4, using
- t-expansion [i] based on digital (155, 211, 531)-net over F4, using
- 11 times m-reduction [i] based on digital (155, 222, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 74, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 74, 177)-net over F64, using
- 11 times m-reduction [i] based on digital (155, 222, 531)-net over F4, using
(211−55, 211, 648)-Net in Base 4 — Constructive
(156, 211, 648)-net in base 4, using
- 41 times duplication [i] based on (155, 210, 648)-net in base 4, using
- trace code for nets [i] based on (15, 70, 216)-net in base 64, using
- base change [i] based on digital (5, 60, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- base change [i] based on digital (5, 60, 216)-net over F128, using
- trace code for nets [i] based on (15, 70, 216)-net in base 64, using
(211−55, 211, 1601)-Net over F4 — Digital
Digital (156, 211, 1601)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4211, 1601, F4, 55) (dual of [1601, 1390, 56]-code), using
- 1389 step Varšamov–Edel lengthening with (ri) = (15, 7, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 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, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 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, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 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, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 26 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 30 times 0, 1, 32 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 37 times 0, 1, 38 times 0) [i] based on linear OA(455, 56, F4, 55) (dual of [56, 1, 56]-code or 56-arc in PG(54,4)), using
- dual of repetition code with length 56 [i]
- 1389 step Varšamov–Edel lengthening with (ri) = (15, 7, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 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, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 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, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 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, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 26 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 30 times 0, 1, 32 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 37 times 0, 1, 38 times 0) [i] based on linear OA(455, 56, F4, 55) (dual of [56, 1, 56]-code or 56-arc in PG(54,4)), using
(211−55, 211, 175355)-Net in Base 4 — Upper bound on s
There is no (156, 211, 175356)-net in base 4, because
- 1 times m-reduction [i] would yield (156, 210, 175356)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 707815 495564 801880 758841 821981 781136 602360 654832 058409 082962 502641 231652 298269 076998 845716 947276 046397 209535 353944 520788 927401 > 4210 [i]