Best Known (69−13, 69, s)-Nets in Base 16
(69−13, 69, 174796)-Net over F16 — Constructive and digital
Digital (56, 69, 174796)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (48, 61, 174763)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 174763, F16, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 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(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
- net defined by OOA [i] based on linear OOA(1661, 174763, F16, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
- digital (2, 8, 33)-net over F16, using
(69−13, 69, 349527)-Net in Base 16 — Constructive
(56, 69, 349527)-net in base 16, using
- net defined by OOA [i] based on OOA(1669, 349527, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1669, 2097163, S16, 13), using
- discarding parts of the base [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on OA(1669, 2097163, S16, 13), using
(69−13, 69, 2957746)-Net over F16 — Digital
Digital (56, 69, 2957746)-net over F16, using
(69−13, 69, large)-Net in Base 16 — Upper bound on s
There is no (56, 69, large)-net in base 16, because
- 11 times m-reduction [i] would yield (56, 58, large)-net in base 16, but