Best Known (65−12, 65, s)-Nets in Base 16
(65−12, 65, 174801)-Net over F16 — Constructive and digital
Digital (53, 65, 174801)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 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 (44, 56, 174763)-net over F16, using
- net defined by OOA [i] based on linear OOA(1656, 174763, F16, 12, 12) (dual of [(174763, 12), 2097100, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1656, 1048578, F16, 12) (dual of [1048578, 1048522, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1656, 1048581, F16, 12) (dual of [1048581, 1048525, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(1651, 1048576, F16, 11) (dual of [1048576, 1048525, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(1656, 1048581, F16, 12) (dual of [1048581, 1048525, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1656, 1048578, F16, 12) (dual of [1048578, 1048522, 13]-code), using
- net defined by OOA [i] based on linear OOA(1656, 174763, F16, 12, 12) (dual of [(174763, 12), 2097100, 13]-NRT-code), using
- digital (3, 9, 38)-net over F16, using
(65−12, 65, 349527)-Net in Base 16 — Constructive
(53, 65, 349527)-net in base 16, using
- 162 times duplication [i] based on (51, 63, 349527)-net in base 16, using
- base change [i] based on digital (24, 36, 349527)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 349527, F128, 12, 12) (dual of [(349527, 12), 4194288, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- 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(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- net defined by OOA [i] based on linear OOA(12836, 349527, F128, 12, 12) (dual of [(349527, 12), 4194288, 13]-NRT-code), using
- base change [i] based on digital (24, 36, 349527)-net over F128, using
(65−12, 65, 4267543)-Net over F16 — Digital
Digital (53, 65, 4267543)-net over F16, using
(65−12, 65, large)-Net in Base 16 — Upper bound on s
There is no (53, 65, large)-net in base 16, because
- 10 times m-reduction [i] would yield (53, 55, large)-net in base 16, but