Best Known (126−24, 126, s)-Nets in Base 16
(126−24, 126, 87419)-Net over F16 — Constructive and digital
Digital (102, 126, 87419)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (87, 111, 87381)-net over F16, using
- net defined by OOA [i] based on linear OOA(16111, 87381, F16, 24, 24) (dual of [(87381, 24), 2097033, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(16111, 1048572, F16, 24) (dual of [1048572, 1048461, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(16111, 1048576, F16, 24) (dual of [1048576, 1048465, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(16111, 1048576, F16, 24) (dual of [1048576, 1048465, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(16111, 1048572, F16, 24) (dual of [1048572, 1048461, 25]-code), using
- net defined by OOA [i] based on linear OOA(16111, 87381, F16, 24, 24) (dual of [(87381, 24), 2097033, 25]-NRT-code), using
- digital (3, 15, 38)-net over F16, using
(126−24, 126, 174763)-Net in Base 16 — Constructive
(102, 126, 174763)-net in base 16, using
- base change [i] based on digital (48, 72, 174763)-net over F128, using
- 1281 times duplication [i] based on digital (47, 71, 174763)-net over F128, using
- net defined by OOA [i] based on linear OOA(12871, 174763, F128, 24, 24) (dual of [(174763, 24), 4194241, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12871, 2097156, F128, 24) (dual of [2097156, 2097085, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12871, 2097156, F128, 24) (dual of [2097156, 2097085, 25]-code), using
- net defined by OOA [i] based on linear OOA(12871, 174763, F128, 24, 24) (dual of [(174763, 24), 4194241, 25]-NRT-code), using
- 1281 times duplication [i] based on digital (47, 71, 174763)-net over F128, using
(126−24, 126, 2482278)-Net over F16 — Digital
Digital (102, 126, 2482278)-net over F16, using
(126−24, 126, large)-Net in Base 16 — Upper bound on s
There is no (102, 126, large)-net in base 16, because
- 22 times m-reduction [i] would yield (102, 104, large)-net in base 16, but