Best Known (159−47, 159, s)-Nets in Base 8
(159−47, 159, 1026)-Net over F8 — Constructive and digital
Digital (112, 159, 1026)-net over F8, using
- 9 times m-reduction [i] based on digital (112, 168, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 84, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 84, 513)-net over F64, using
(159−47, 159, 3425)-Net over F8 — Digital
Digital (112, 159, 3425)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 3425, F8, 47) (dual of [3425, 3266, 48]-code), using
- 3265 step Varšamov–Edel lengthening with (ri) = (8, 3, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 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, 0, 1, 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, 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, 7 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, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 47 times 0, 1, 49 times 0, 1, 51 times 0, 1, 53 times 0, 1, 56 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 67 times 0, 1, 71 times 0, 1, 75 times 0, 1, 77 times 0, 1, 82 times 0, 1, 85 times 0, 1, 90 times 0, 1, 93 times 0, 1, 99 times 0, 1, 102 times 0, 1, 108 times 0, 1, 112 times 0, 1, 118 times 0, 1, 124 times 0, 1, 129 times 0, 1, 135 times 0, 1, 142 times 0, 1, 149 times 0) [i] based on linear OA(847, 48, F8, 47) (dual of [48, 1, 48]-code or 48-arc in PG(46,8)), using
- dual of repetition code with length 48 [i]
- 3265 step Varšamov–Edel lengthening with (ri) = (8, 3, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 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, 0, 1, 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, 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, 7 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, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 47 times 0, 1, 49 times 0, 1, 51 times 0, 1, 53 times 0, 1, 56 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 67 times 0, 1, 71 times 0, 1, 75 times 0, 1, 77 times 0, 1, 82 times 0, 1, 85 times 0, 1, 90 times 0, 1, 93 times 0, 1, 99 times 0, 1, 102 times 0, 1, 108 times 0, 1, 112 times 0, 1, 118 times 0, 1, 124 times 0, 1, 129 times 0, 1, 135 times 0, 1, 142 times 0, 1, 149 times 0) [i] based on linear OA(847, 48, F8, 47) (dual of [48, 1, 48]-code or 48-arc in PG(46,8)), using
(159−47, 159, 2153728)-Net in Base 8 — Upper bound on s
There is no (112, 159, 2153729)-net in base 8, because
- 1 times m-reduction [i] would yield (112, 158, 2153729)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 48777 667879 644172 917459 008562 770663 191537 983474 318723 867702 180432 919172 892426 636053 885497 585301 700546 524597 321057 556905 792216 717698 189203 278080 > 8158 [i]