Best Known (86−31, 86, s)-Nets in Base 16
(86−31, 86, 581)-Net over F16 — Constructive and digital
Digital (55, 86, 581)-net over F16, using
- 161 times duplication [i] based on digital (54, 85, 581)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (33, 64, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 32, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 32, 258)-net over F256, using
- digital (6, 21, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(86−31, 86, 594)-Net in Base 16 — Constructive
(55, 86, 594)-net in base 16, using
- (u, u+v)-construction [i] based on
- (9, 24, 80)-net in base 16, using
- base change [i] based on digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 16, 80)-net over F64, using
- digital (31, 62, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 31, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 31, 257)-net over F256, using
- (9, 24, 80)-net in base 16, using
(86−31, 86, 2288)-Net over F16 — Digital
Digital (55, 86, 2288)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1686, 2288, F16, 31) (dual of [2288, 2202, 32]-code), using
- 2201 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 22 times 0, 1, 25 times 0, 1, 26 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 39 times 0, 1, 44 times 0, 1, 48 times 0, 1, 53 times 0, 1, 58 times 0, 1, 65 times 0, 1, 70 times 0, 1, 78 times 0, 1, 85 times 0, 1, 94 times 0, 1, 103 times 0, 1, 113 times 0, 1, 125 times 0, 1, 136 times 0, 1, 150 times 0, 1, 165 times 0, 1, 181 times 0, 1, 199 times 0) [i] based on linear OA(1631, 32, F16, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,16)), using
- dual of repetition code with length 32 [i]
- 2201 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 22 times 0, 1, 25 times 0, 1, 26 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 39 times 0, 1, 44 times 0, 1, 48 times 0, 1, 53 times 0, 1, 58 times 0, 1, 65 times 0, 1, 70 times 0, 1, 78 times 0, 1, 85 times 0, 1, 94 times 0, 1, 103 times 0, 1, 113 times 0, 1, 125 times 0, 1, 136 times 0, 1, 150 times 0, 1, 165 times 0, 1, 181 times 0, 1, 199 times 0) [i] based on linear OA(1631, 32, F16, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,16)), using
(86−31, 86, 2851154)-Net in Base 16 — Upper bound on s
There is no (55, 86, 2851155)-net in base 16, because
- 1 times m-reduction [i] would yield (55, 85, 2851155)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 2 239752 200571 748967 657406 423466 824279 583370 335653 675723 288286 965245 961907 911203 817246 555563 309208 567376 > 1685 [i]