Best Known (68, 68+23, s)-Nets in Base 8
(68, 68+23, 745)-Net over F8 — Constructive and digital
Digital (68, 91, 745)-net over F8, using
- 81 times duplication [i] based on digital (67, 90, 745)-net over F8, using
- net defined by OOA [i] based on linear OOA(890, 745, F8, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- net defined by OOA [i] based on linear OOA(890, 745, F8, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
(68, 68+23, 814)-Net in Base 8 — Constructive
(68, 91, 814)-net in base 8, using
- 81 times duplication [i] based on (67, 90, 814)-net in base 8, using
- (u, u+v)-construction [i] based on
- (17, 28, 300)-net in base 8, using
- trace code for nets [i] based on (3, 14, 150)-net in base 64, using
- base change [i] based on digital (1, 12, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 12, 150)-net over F128, using
- trace code for nets [i] based on (3, 14, 150)-net in base 64, using
- (39, 62, 514)-net in base 8, using
- trace code for nets [i] based on (8, 31, 257)-net in base 64, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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
- base change [i] based on digital (0, 24, 257)-net over F256, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- trace code for nets [i] based on (8, 31, 257)-net in base 64, using
- (17, 28, 300)-net in base 8, using
- (u, u+v)-construction [i] based on
(68, 68+23, 8198)-Net over F8 — Digital
Digital (68, 91, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(891, 8198, F8, 23) (dual of [8198, 8107, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- linear OA(890, 8197, F8, 22) (dual of [8197, 8107, 23]-code), using Gilbert–Varšamov bound and bm = 890 > Vbs−1(k−1) = 163 447529 669905 109386 132904 328472 346939 695133 140897 651741 741359 952971 529079 625728 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- construction X with Varšamov bound [i] based on
(68, 68+23, large)-Net in Base 8 — Upper bound on s
There is no (68, 91, large)-net in base 8, because
- 21 times m-reduction [i] would yield (68, 70, large)-net in base 8, but