Best Known (86, 86+25, s)-Nets in Base 16
(86, 86+25, 10946)-Net over F16 — Constructive and digital
Digital (86, 111, 10946)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (73, 98, 10922)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 10922, F16, 25, 25) (dual of [(10922, 25), 272952, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1698, 131065, F16, 25) (dual of [131065, 130967, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 131074, F16, 25) (dual of [131074, 130976, 26]-code), using
- trace code [i] based on linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- trace code [i] based on linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 131074, F16, 25) (dual of [131074, 130976, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1698, 131065, F16, 25) (dual of [131065, 130967, 26]-code), using
- net defined by OOA [i] based on linear OOA(1698, 10922, F16, 25, 25) (dual of [(10922, 25), 272952, 26]-NRT-code), using
- digital (1, 13, 24)-net over F16, using
(86, 86+25, 21845)-Net in Base 16 — Constructive
(86, 111, 21845)-net in base 16, using
- base change [i] based on digital (49, 74, 21845)-net over F64, using
- 641 times duplication [i] based on digital (48, 73, 21845)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
- 641 times duplication [i] based on digital (48, 73, 21845)-net over F64, using
(86, 86+25, 242297)-Net over F16 — Digital
Digital (86, 111, 242297)-net over F16, using
(86, 86+25, large)-Net in Base 16 — Upper bound on s
There is no (86, 111, large)-net in base 16, because
- 23 times m-reduction [i] would yield (86, 88, large)-net in base 16, but