Best Known (65, 65+27, s)-Nets in Base 8
(65, 65+27, 416)-Net over F8 — Constructive and digital
Digital (65, 92, 416)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (19, 32, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- digital (33, 60, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 30, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64 (see above)
- trace code for nets [i] based on digital (3, 30, 104)-net over F64, using
- digital (19, 32, 208)-net over F8, using
(65, 65+27, 576)-Net in Base 8 — Constructive
(65, 92, 576)-net in base 8, using
- 6 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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, 42, 288)-net over F128, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
(65, 65+27, 2378)-Net over F8 — Digital
Digital (65, 92, 2378)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(892, 2378, F8, 27) (dual of [2378, 2286, 28]-code), using
- 2285 step Varšamov–Edel lengthening with (ri) = (5, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 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, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 37 times 0, 1, 41 times 0, 1, 45 times 0, 1, 49 times 0, 1, 52 times 0, 1, 58 times 0, 1, 62 times 0, 1, 68 times 0, 1, 73 times 0, 1, 80 times 0, 1, 86 times 0, 1, 94 times 0, 1, 102 times 0, 1, 110 times 0, 1, 120 times 0, 1, 130 times 0, 1, 141 times 0, 1, 153 times 0, 1, 166 times 0, 1, 180 times 0) [i] based on linear OA(827, 28, F8, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,8)), using
- dual of repetition code with length 28 [i]
- 2285 step Varšamov–Edel lengthening with (ri) = (5, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 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, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 37 times 0, 1, 41 times 0, 1, 45 times 0, 1, 49 times 0, 1, 52 times 0, 1, 58 times 0, 1, 62 times 0, 1, 68 times 0, 1, 73 times 0, 1, 80 times 0, 1, 86 times 0, 1, 94 times 0, 1, 102 times 0, 1, 110 times 0, 1, 120 times 0, 1, 130 times 0, 1, 141 times 0, 1, 153 times 0, 1, 166 times 0, 1, 180 times 0) [i] based on linear OA(827, 28, F8, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,8)), using
(65, 65+27, 1697993)-Net in Base 8 — Upper bound on s
There is no (65, 92, 1697994)-net in base 8, because
- 1 times m-reduction [i] would yield (65, 91, 1697994)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 15177 152415 880119 714807 438609 318155 164390 867711 427324 174017 586347 092505 924271 512080 > 891 [i]