Best Known (78, 78+26, s)-Nets in Base 8
(78, 78+26, 630)-Net over F8 — Constructive and digital
Digital (78, 104, 630)-net over F8, using
- 82 times duplication [i] based on digital (76, 102, 630)-net over F8, using
- net defined by OOA [i] based on linear OOA(8102, 630, F8, 26, 26) (dual of [(630, 26), 16278, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8102, 8190, F8, 26) (dual of [8190, 8088, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8102, 8192, F8, 26) (dual of [8192, 8090, 27]-code), using
- trace code [i] based on linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- trace code [i] based on linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8102, 8192, F8, 26) (dual of [8192, 8090, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8102, 8190, F8, 26) (dual of [8190, 8088, 27]-code), using
- net defined by OOA [i] based on linear OOA(8102, 630, F8, 26, 26) (dual of [(630, 26), 16278, 27]-NRT-code), using
(78, 78+26, 1028)-Net in Base 8 — Constructive
(78, 104, 1028)-net in base 8, using
- base change [i] based on digital (52, 78, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 257)-net over F256, using
- digital (26, 52, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 26, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 26, 257)-net over F256, using
- digital (13, 26, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(78, 78+26, 8318)-Net over F8 — Digital
Digital (78, 104, 8318)-net over F8, using
(78, 78+26, large)-Net in Base 8 — Upper bound on s
There is no (78, 104, large)-net in base 8, because
- 24 times m-reduction [i] would yield (78, 80, large)-net in base 8, but