Best Known (41, 41+16, s)-Nets in Base 8
(41, 41+16, 512)-Net over F8 — Constructive and digital
Digital (41, 57, 512)-net over F8, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(857, 512, F8, 17, 17) (dual of [(512, 17), 8647, 18]-NRT-code), using
(41, 41+16, 547)-Net in Base 8 — Constructive
(41, 57, 547)-net in base 8, using
- 81 times duplication [i] based on (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
- base change [i] based on digital (26, 42, 547)-net over F16, using
(41, 41+16, 3530)-Net over F8 — Digital
Digital (41, 57, 3530)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 3530, F8, 16) (dual of [3530, 3473, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 4100, F8, 16) (dual of [4100, 4043, 17]-code), using
- 1 times truncation [i] based on linear OA(858, 4101, F8, 17) (dual of [4101, 4043, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] 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]
- linear OA(853, 4096, F8, 15) (dual of [4096, 4043, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(81, 5, F8, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(858, 4101, F8, 17) (dual of [4101, 4043, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 4100, F8, 16) (dual of [4100, 4043, 17]-code), using
(41, 41+16, 1462536)-Net in Base 8 — Upper bound on s
There is no (41, 57, 1462537)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2993 158542 072458 233028 360699 161438 234359 409366 514977 > 857 [i]