Best Known (219−51, 219, s)-Nets in Base 4
(219−51, 219, 1040)-Net over F4 — Constructive and digital
Digital (168, 219, 1040)-net over F4, using
- 1 times m-reduction [i] based on digital (168, 220, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 55, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 55, 260)-net over F256, using
(219−51, 219, 2841)-Net over F4 — Digital
Digital (168, 219, 2841)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4219, 2841, F4, 51) (dual of [2841, 2622, 52]-code), using
- 2621 step Varšamov–Edel lengthening with (ri) = (14, 6, 4, 2, 2, 2, 1, 2, 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, 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, 1, 0, 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, 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, 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, 7 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, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 9 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, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 29 times 0, 1, 31 times 0, 1, 31 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 times 0, 1, 37 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 42 times 0, 1, 43 times 0, 1, 45 times 0, 1, 46 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 58 times 0, 1, 59 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0, 1, 75 times 0) [i] based on linear OA(451, 52, F4, 51) (dual of [52, 1, 52]-code or 52-arc in PG(50,4)), using
- dual of repetition code with length 52 [i]
- 2621 step Varšamov–Edel lengthening with (ri) = (14, 6, 4, 2, 2, 2, 1, 2, 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, 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, 1, 0, 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, 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, 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, 7 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, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 9 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, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 29 times 0, 1, 31 times 0, 1, 31 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 times 0, 1, 37 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 42 times 0, 1, 43 times 0, 1, 45 times 0, 1, 46 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 58 times 0, 1, 59 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0, 1, 75 times 0) [i] based on linear OA(451, 52, F4, 51) (dual of [52, 1, 52]-code or 52-arc in PG(50,4)), using
(219−51, 219, 603188)-Net in Base 4 — Upper bound on s
There is no (168, 219, 603189)-net in base 4, because
- 1 times m-reduction [i] would yield (168, 218, 603189)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 177454 493476 476696 235921 938387 670418 192607 904597 804358 959740 353825 700604 884263 536225 640536 251316 435415 400084 250938 048777 197139 303520 > 4218 [i]