Best Known (54, 54+12, s)-Nets in Base 16
(54, 54+12, 174808)-Net over F16 — Constructive and digital
Digital (54, 66, 174808)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-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 (4, 10, 45)-net over F16, using
(54, 54+12, 349542)-Net in Base 16 — Constructive
(54, 66, 349542)-net in base 16, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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
- (48, 60, 349525)-net in base 16, using
- net defined by OOA [i] based on OOA(1660, 349525, S16, 12, 12), using
- OA 6-folding and stacking [i] based on OA(1660, 2097150, S16, 12), using
- discarding factors based on OA(1660, 2097155, S16, 12), using
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [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(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- discarding factors based on OA(1660, 2097155, S16, 12), using
- OA 6-folding and stacking [i] based on OA(1660, 2097150, S16, 12), using
- net defined by OOA [i] based on OOA(1660, 349525, S16, 12, 12), using
- digital (0, 6, 17)-net over F16, using
(54, 54+12, 5490896)-Net over F16 — Digital
Digital (54, 66, 5490896)-net over F16, using
(54, 54+12, large)-Net in Base 16 — Upper bound on s
There is no (54, 66, large)-net in base 16, because
- 10 times m-reduction [i] would yield (54, 56, large)-net in base 16, but