Best Known (65, 65+22, s)-Nets in Base 8
(65, 65+22, 745)-Net over F8 — Constructive and digital
Digital (65, 87, 745)-net over F8, using
- 81 times duplication [i] based on digital (64, 86, 745)-net over F8, using
- net defined by OOA [i] based on linear OOA(886, 745, F8, 22, 22) (dual of [(745, 22), 16304, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(886, 8195, F8, 22) (dual of [8195, 8109, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(886, 8195, F8, 22) (dual of [8195, 8109, 23]-code), using
- net defined by OOA [i] based on linear OOA(886, 745, F8, 22, 22) (dual of [(745, 22), 16304, 23]-NRT-code), using
(65, 65+22, 772)-Net in Base 8 — Constructive
(65, 87, 772)-net in base 8, using
- 1 times m-reduction [i] based on (65, 88, 772)-net in base 8, using
- (u, u+v)-construction [i] based on
- (15, 26, 258)-net in base 8, using
- trace code for nets [i] based on (2, 13, 129)-net in base 64, using
- 1 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
- 1 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
- trace code for nets [i] based on (2, 13, 129)-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
- (15, 26, 258)-net in base 8, using
- (u, u+v)-construction [i] based on
(65, 65+22, 8198)-Net over F8 — Digital
Digital (65, 87, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(887, 8198, F8, 22) (dual of [8198, 8111, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- linear OA(886, 8197, F8, 21) (dual of [8197, 8111, 22]-code), using Gilbert–Varšamov bound and bm = 886 > Vbs−1(k−1) = 59972 357824 589686 329208 835458 923368 646940 052636 233577 069583 570138 295988 074496 [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(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- construction X with Varšamov bound [i] based on
(65, 65+22, large)-Net in Base 8 — Upper bound on s
There is no (65, 87, large)-net in base 8, because
- 20 times m-reduction [i] would yield (65, 67, large)-net in base 8, but