Best Known (109−21, 109, s)-Nets in Base 16
(109−21, 109, 104896)-Net over F16 — Constructive and digital
Digital (88, 109, 104896)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 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 (75, 96, 104858)-net over F16, using
- net defined by OOA [i] based on linear OOA(1696, 104858, F16, 21, 21) (dual of [(104858, 21), 2201922, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1696, 1048581, F16, 21) (dual of [1048581, 1048485, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(1696, 1048581, F16, 21) (dual of [1048581, 1048485, 22]-code), using
- net defined by OOA [i] based on linear OOA(1696, 104858, F16, 21, 21) (dual of [(104858, 21), 2201922, 22]-NRT-code), using
- digital (3, 13, 38)-net over F16, using
(109−21, 109, 209715)-Net in Base 16 — Constructive
(88, 109, 209715)-net in base 16, using
- 162 times duplication [i] based on (86, 107, 209715)-net in base 16, using
- net defined by OOA [i] based on OOA(16107, 209715, S16, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(16107, 2097151, S16, 21), using
- discarding factors based on OA(16107, 2097155, S16, 21), using
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- discarding factors based on OA(16107, 2097155, S16, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(16107, 2097151, S16, 21), using
- net defined by OOA [i] based on OOA(16107, 209715, S16, 21, 21), using
(109−21, 109, 2021487)-Net over F16 — Digital
Digital (88, 109, 2021487)-net over F16, using
(109−21, 109, large)-Net in Base 16 — Upper bound on s
There is no (88, 109, large)-net in base 16, because
- 19 times m-reduction [i] would yield (88, 90, large)-net in base 16, but