Best Known (109, 109+33, s)-Nets in Base 4
(109, 109+33, 1036)-Net over F4 — Constructive and digital
Digital (109, 142, 1036)-net over F4, using
- 42 times duplication [i] based on digital (107, 140, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 35, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 35, 259)-net over F256, using
(109, 109+33, 2018)-Net over F4 — Digital
Digital (109, 142, 2018)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4142, 2018, F4, 33) (dual of [2018, 1876, 34]-code), using
- 1875 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, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 41 times 0, 1, 42 times 0, 1, 44 times 0, 1, 47 times 0, 1, 48 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 58 times 0, 1, 61 times 0, 1, 63 times 0, 1, 67 times 0, 1, 69 times 0, 1, 73 times 0, 1, 76 times 0, 1, 79 times 0, 1, 83 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]
- 1875 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, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 41 times 0, 1, 42 times 0, 1, 44 times 0, 1, 47 times 0, 1, 48 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 58 times 0, 1, 61 times 0, 1, 63 times 0, 1, 67 times 0, 1, 69 times 0, 1, 73 times 0, 1, 76 times 0, 1, 79 times 0, 1, 83 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
(109, 109+33, 458204)-Net in Base 4 — Upper bound on s
There is no (109, 142, 458205)-net in base 4, because
- 1 times m-reduction [i] would yield (109, 141, 458205)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 770917 150128 491524 491672 380292 392858 068765 922931 566329 162313 818740 369318 472042 905159 > 4141 [i]