Best Known (75, 75+31, s)-Nets in Base 8
(75, 75+31, 484)-Net over F8 — Constructive and digital
Digital (75, 106, 484)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (15, 30, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 15, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 15, 65)-net over F64, using
- digital (45, 76, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 177)-net over F64, using
- digital (15, 30, 130)-net over F8, using
(75, 75+31, 576)-Net in Base 8 — Constructive
(75, 106, 576)-net in base 8, using
- t-expansion [i] based on (73, 106, 576)-net in base 8, using
- 6 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
- 6 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
(75, 75+31, 2686)-Net over F8 — Digital
Digital (75, 106, 2686)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8106, 2686, F8, 31) (dual of [2686, 2580, 32]-code), using
- 2579 step Varšamov–Edel lengthening with (ri) = (5, 3, 1, 1, 1, 1, 1, 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, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 37 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 49 times 0, 1, 53 times 0, 1, 57 times 0, 1, 61 times 0, 1, 66 times 0, 1, 71 times 0, 1, 75 times 0, 1, 82 times 0, 1, 87 times 0, 1, 94 times 0, 1, 101 times 0, 1, 108 times 0, 1, 116 times 0, 1, 124 times 0, 1, 134 times 0, 1, 143 times 0, 1, 154 times 0, 1, 165 times 0, 1, 177 times 0) [i] based on linear OA(831, 32, F8, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,8)), using
- dual of repetition code with length 32 [i]
- 2579 step Varšamov–Edel lengthening with (ri) = (5, 3, 1, 1, 1, 1, 1, 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, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 37 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 49 times 0, 1, 53 times 0, 1, 57 times 0, 1, 61 times 0, 1, 66 times 0, 1, 71 times 0, 1, 75 times 0, 1, 82 times 0, 1, 87 times 0, 1, 94 times 0, 1, 101 times 0, 1, 108 times 0, 1, 116 times 0, 1, 124 times 0, 1, 134 times 0, 1, 143 times 0, 1, 154 times 0, 1, 165 times 0, 1, 177 times 0) [i] based on linear OA(831, 32, F8, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,8)), using
(75, 75+31, 1924404)-Net in Base 8 — Upper bound on s
There is no (75, 106, 1924405)-net in base 8, because
- 1 times m-reduction [i] would yield (75, 105, 1924405)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 66749 664669 073053 184648 224466 179945 094725 126939 985840 536831 477185 639970 457369 412072 986472 366672 > 8105 [i]