Best Known (87−25, 87, s)-Nets in Base 8
(87−25, 87, 419)-Net over F8 — Constructive and digital
Digital (62, 87, 419)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (11, 23, 65)-net over F8, using
- base reduction for projective spaces (embedding PG(11,64) in PG(22,8)) for nets [i] based on digital (0, 12, 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
- base reduction for projective spaces (embedding PG(11,64) in PG(22,8)) for nets [i] based on digital (0, 12, 65)-net over F64, using
- digital (39, 64, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 32, 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, 32, 177)-net over F64, using
- digital (11, 23, 65)-net over F8, using
(87−25, 87, 576)-Net in Base 8 — Constructive
(62, 87, 576)-net in base 8, using
- 5 times m-reduction [i] based on (62, 92, 576)-net in base 8, using
- trace code for nets [i] based on (16, 46, 288)-net in base 64, using
- 3 times m-reduction [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
- 3 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- trace code for nets [i] based on (16, 46, 288)-net in base 64, using
(87−25, 87, 3194)-Net over F8 — Digital
Digital (62, 87, 3194)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(887, 3194, F8, 25) (dual of [3194, 3107, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(887, 4105, F8, 25) (dual of [4105, 4018, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(885, 4096, F8, 25) (dual of [4096, 4011, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(887, 4105, F8, 25) (dual of [4105, 4018, 26]-code), using
(87−25, 87, 2240819)-Net in Base 8 — Upper bound on s
There is no (62, 87, 2240820)-net in base 8, because
- 1 times m-reduction [i] would yield (62, 86, 2240820)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 463169 506309 548272 222962 506177 945138 099415 930181 879916 741810 898554 990133 641932 > 886 [i]