Best Known (40, 57, s)-Nets in Base 8
(40, 57, 512)-Net over F8 — Constructive and digital
Digital (40, 57, 512)-net over F8, using
- net defined by OOA [i] based on linear OOA(857, 512, F8, 17, 17) (dual of [(512, 17), 8647, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
(40, 57, 538)-Net in Base 8 — Constructive
(40, 57, 538)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- (29, 46, 514)-net in base 8, using
- trace code for nets [i] based on (6, 23, 257)-net in base 64, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- trace code for nets [i] based on (6, 23, 257)-net in base 64, using
- digital (3, 11, 24)-net over F8, using
(40, 57, 2150)-Net over F8 — Digital
Digital (40, 57, 2150)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 2150, F8, 17) (dual of [2150, 2093, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using
(40, 57, 1127768)-Net in Base 8 — Upper bound on s
There is no (40, 57, 1127769)-net in base 8, because
- 1 times m-reduction [i] would yield (40, 56, 1127769)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 374 144461 434096 067784 717932 918782 955461 487885 649255 > 856 [i]