Best Known (126−41, 126, s)-Nets in Base 8
(126−41, 126, 400)-Net over F8 — Constructive and digital
Digital (85, 126, 400)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (10, 30, 46)-net over F8, using
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 9, N(F) = 45, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
- digital (55, 96, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 177)-net over F64, using
- digital (10, 30, 46)-net over F8, using
(126−41, 126, 576)-Net in Base 8 — Constructive
(85, 126, 576)-net in base 8, using
- 6 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
- 4 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
- 4 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
(126−41, 126, 1596)-Net over F8 — Digital
Digital (85, 126, 1596)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8126, 1596, F8, 41) (dual of [1596, 1470, 42]-code), using
- 1469 step Varšamov–Edel lengthening with (ri) = (7, 3, 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, 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, 6 times 0, 1, 5 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, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 48 times 0, 1, 50 times 0, 1, 54 times 0, 1, 56 times 0, 1, 60 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 74 times 0, 1, 78 times 0) [i] based on linear OA(841, 42, F8, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,8)), using
- dual of repetition code with length 42 [i]
- 1469 step Varšamov–Edel lengthening with (ri) = (7, 3, 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, 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, 6 times 0, 1, 5 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, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 48 times 0, 1, 50 times 0, 1, 54 times 0, 1, 56 times 0, 1, 60 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 74 times 0, 1, 78 times 0) [i] based on linear OA(841, 42, F8, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,8)), using
(126−41, 126, 523010)-Net in Base 8 — Upper bound on s
There is no (85, 126, 523011)-net in base 8, because
- 1 times m-reduction [i] would yield (85, 125, 523011)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 76958 724610 144310 514769 458949 954419 510756 456332 831621 846857 654682 927389 983416 548028 342108 215238 927190 456123 077824 > 8125 [i]