Best Known (65−19, 65, s)-Nets in Base 8
(65−19, 65, 455)-Net over F8 — Constructive and digital
Digital (46, 65, 455)-net over F8, using
- net defined by OOA [i] based on linear OOA(865, 455, F8, 19, 19) (dual of [(455, 19), 8580, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(865, 4096, F8, 19) (dual of [4096, 4031, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(865, 4096, F8, 19) (dual of [4096, 4031, 20]-code), using
(65−19, 65, 539)-Net in Base 8 — Constructive
(46, 65, 539)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- (33, 52, 514)-net in base 8, using
- base change [i] based on digital (20, 39, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (20, 40, 514)-net over F16, using
- base change [i] based on digital (20, 39, 514)-net over F16, using
- digital (4, 13, 25)-net over F8, using
(65−19, 65, 2565)-Net over F8 — Digital
Digital (46, 65, 2565)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(865, 2565, F8, 19) (dual of [2565, 2500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(865, 4096, F8, 19) (dual of [4096, 4031, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(865, 4096, F8, 19) (dual of [4096, 4031, 20]-code), using
(65−19, 65, 1565399)-Net in Base 8 — Upper bound on s
There is no (46, 65, 1565400)-net in base 8, because
- 1 times m-reduction [i] would yield (46, 64, 1565400)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 6277 120857 409275 648785 898545 627460 691750 640127 730650 868786 > 864 [i]