Best Known (44, 62, s)-Nets in Base 8
(44, 62, 455)-Net over F8 — Constructive and digital
Digital (44, 62, 455)-net over F8, using
- 81 times duplication [i] based on digital (43, 61, 455)-net over F8, using
- net defined by OOA [i] based on linear OOA(861, 455, F8, 18, 18) (dual of [(455, 18), 8129, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(861, 4095, F8, 18) (dual of [4095, 4034, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(861, 4095, F8, 18) (dual of [4095, 4034, 19]-code), using
- net defined by OOA [i] based on linear OOA(861, 455, F8, 18, 18) (dual of [(455, 18), 8129, 19]-NRT-code), using
(44, 62, 542)-Net in Base 8 — Constructive
(44, 62, 542)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- (30, 48, 514)-net in base 8, using
- base change [i] based on digital (18, 36, 514)-net over F16, using
- trace code for nets [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
- trace code for nets [i] based on digital (0, 18, 257)-net over F256, using
- base change [i] based on digital (18, 36, 514)-net over F16, using
- digital (5, 14, 28)-net over F8, using
(44, 62, 2685)-Net over F8 — Digital
Digital (44, 62, 2685)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(862, 2685, F8, 18) (dual of [2685, 2623, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 4101, F8, 18) (dual of [4101, 4039, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(861, 4100, F8, 18) (dual of [4100, 4039, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(861, 4100, F8, 18) (dual of [4100, 4039, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 4101, F8, 18) (dual of [4101, 4039, 19]-code), using
(44, 62, 986137)-Net in Base 8 — Upper bound on s
There is no (44, 62, 986138)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 98 079789 763732 370080 367707 965765 704931 320771 052476 019153 > 862 [i]