Best Known (67, 67+26, s)-Nets in Base 8
(67, 67+26, 484)-Net over F8 — Constructive and digital
Digital (67, 93, 484)-net over F8, using
- 1 times m-reduction [i] based on digital (67, 94, 484)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 65)-net over F64, using
- digital (41, 68, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 177)-net over F64, using
- digital (13, 26, 130)-net over F8, using
- (u, u+v)-construction [i] based on
(67, 67+26, 576)-Net in Base 8 — Constructive
(67, 93, 576)-net in base 8, using
- t-expansion [i] based on (65, 93, 576)-net in base 8, using
- 5 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
- 5 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
(67, 67+26, 4042)-Net over F8 — Digital
Digital (67, 93, 4042)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(893, 4042, F8, 26) (dual of [4042, 3949, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(893, 4112, F8, 26) (dual of [4112, 4019, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(889, 4096, F8, 26) (dual of [4096, 4007, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(877, 4096, F8, 22) (dual of [4096, 4019, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(84, 16, F8, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,8)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(893, 4112, F8, 26) (dual of [4112, 4019, 27]-code), using
(67, 67+26, 2338156)-Net in Base 8 — Upper bound on s
There is no (67, 93, 2338157)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 971339 114041 507515 808700 951682 339734 036586 177521 609542 256809 532259 784180 019055 400296 > 893 [i]