Best Known (59, 59+14, s)-Nets in Base 16
(59, 59+14, 149814)-Net over F16 — Constructive and digital
Digital (59, 73, 149814)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (52, 66, 149797)-net over F16, using
- net defined by OOA [i] based on linear OOA(1666, 149797, F16, 14, 14) (dual of [(149797, 14), 2097092, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1666, 1048579, F16, 14) (dual of [1048579, 1048513, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1666, 1048581, F16, 14) (dual of [1048581, 1048515, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(1666, 1048581, F16, 14) (dual of [1048581, 1048515, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1666, 1048579, F16, 14) (dual of [1048579, 1048513, 15]-code), using
- net defined by OOA [i] based on linear OOA(1666, 149797, F16, 14, 14) (dual of [(149797, 14), 2097092, 15]-NRT-code), using
- digital (0, 7, 17)-net over F16, using
(59, 59+14, 299594)-Net in Base 16 — Constructive
(59, 73, 299594)-net in base 16, using
- 161 times duplication [i] based on (58, 72, 299594)-net in base 16, using
- net defined by OOA [i] based on OOA(1672, 299594, S16, 14, 14), using
- OA 7-folding and stacking [i] based on OA(1672, 2097158, S16, 14), using
- discarding factors based on OA(1672, 2097159, S16, 14), using
- discarding parts of the base [i] based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- discarding factors based on OA(1672, 2097159, S16, 14), using
- OA 7-folding and stacking [i] based on OA(1672, 2097158, S16, 14), using
- net defined by OOA [i] based on OOA(1672, 299594, S16, 14, 14), using
(59, 59+14, 2182293)-Net over F16 — Digital
Digital (59, 73, 2182293)-net over F16, using
(59, 59+14, large)-Net in Base 16 — Upper bound on s
There is no (59, 73, large)-net in base 16, because
- 12 times m-reduction [i] would yield (59, 61, large)-net in base 16, but