Best Known (135−31, 135, s)-Nets in Base 4
(135−31, 135, 1036)-Net over F4 — Constructive and digital
Digital (104, 135, 1036)-net over F4, using
- 1 times m-reduction [i] based on digital (104, 136, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 34, 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, 34, 259)-net over F256, using
(135−31, 135, 2071)-Net over F4 — Digital
Digital (104, 135, 2071)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4135, 2071, F4, 31) (dual of [2071, 1936, 32]-code), using
- 1935 step Varšamov–Edel lengthening with (ri) = (9, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 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, 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, 5 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, 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, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0) [i] based on linear OA(431, 32, F4, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,4)), using
- dual of repetition code with length 32 [i]
- 1935 step Varšamov–Edel lengthening with (ri) = (9, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 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, 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, 5 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, 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, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0) [i] based on linear OA(431, 32, F4, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,4)), using
(135−31, 135, 511726)-Net in Base 4 — Upper bound on s
There is no (104, 135, 511727)-net in base 4, because
- 1 times m-reduction [i] would yield (104, 134, 511727)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 474 293256 869678 645403 305393 421698 081211 090490 058343 226650 152540 842547 156941 950124 > 4134 [i]