Best Known (40, 40+16, s)-Nets in Base 8
(40, 40+16, 512)-Net over F8 — Constructive and digital
Digital (40, 56, 512)-net over F8, using
- net defined by OOA [i] based on linear OOA(856, 512, F8, 16, 16) (dual of [(512, 16), 8136, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
(40, 40+16, 547)-Net in Base 8 — Constructive
(40, 56, 547)-net in base 8, using
- base change [i] based on digital (26, 42, 547)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (16, 32, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (2, 10, 33)-net over F16, using
- (u, u+v)-construction [i] based on
(40, 40+16, 3041)-Net over F8 — Digital
Digital (40, 56, 3041)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 3041, F8, 16) (dual of [3041, 2985, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
(40, 40+16, 1127768)-Net in Base 8 — Upper bound on s
There is no (40, 56, 1127769)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 374 144461 434096 067784 717932 918782 955461 487885 649255 > 856 [i]