Best Known (85, 85+25, s)-Nets in Base 16
(85, 85+25, 10939)-Net over F16 — Constructive and digital
Digital (85, 110, 10939)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-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 (0, 12, 17)-net over F16, using
(85, 85+25, 21845)-Net in Base 16 — Constructive
(85, 110, 21845)-net in base 16, using
- net defined by OOA [i] based on OOA(16110, 21845, S16, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(16110, 262141, S16, 25), using
- discarding factors based on OA(16110, 262147, S16, 25), using
- discarding parts of the base [i] based on linear OA(6473, 262147, F64, 25) (dual of [262147, 262074, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] 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]
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(6473, 262147, F64, 25) (dual of [262147, 262074, 26]-code), using
- discarding factors based on OA(16110, 262147, S16, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(16110, 262141, S16, 25), using
(85, 85+25, 215863)-Net over F16 — Digital
Digital (85, 110, 215863)-net over F16, using
(85, 85+25, large)-Net in Base 16 — Upper bound on s
There is no (85, 110, large)-net in base 16, because
- 23 times m-reduction [i] would yield (85, 87, large)-net in base 16, but