Best Known (51−10, 51, s)-Nets in Base 16
(51−10, 51, 209733)-Net over F16 — Constructive and digital
Digital (41, 51, 209733)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (36, 46, 209716)-net over F16, using
- net defined by OOA [i] based on linear OOA(1646, 209716, F16, 10, 10) (dual of [(209716, 10), 2097114, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1646, 1048580, F16, 10) (dual of [1048580, 1048534, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1646, 1048581, F16, 10) (dual of [1048581, 1048535, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1641, 1048576, F16, 9) (dual of [1048576, 1048535, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(1646, 1048581, F16, 10) (dual of [1048581, 1048535, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1646, 1048580, F16, 10) (dual of [1048580, 1048534, 11]-code), using
- net defined by OOA [i] based on linear OOA(1646, 209716, F16, 10, 10) (dual of [(209716, 10), 2097114, 11]-NRT-code), using
- digital (0, 5, 17)-net over F16, using
(51−10, 51, 419431)-Net in Base 16 — Constructive
(41, 51, 419431)-net in base 16, using
- 162 times duplication [i] based on (39, 49, 419431)-net in base 16, using
- base change [i] based on digital (18, 28, 419431)-net over F128, using
- net defined by OOA [i] based on linear OOA(12828, 419431, F128, 10, 10) (dual of [(419431, 10), 4194282, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- 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(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(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(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- net defined by OOA [i] based on linear OOA(12828, 419431, F128, 10, 10) (dual of [(419431, 10), 4194282, 11]-NRT-code), using
- base change [i] based on digital (18, 28, 419431)-net over F128, using
(51−10, 51, 1840805)-Net over F16 — Digital
Digital (41, 51, 1840805)-net over F16, using
(51−10, 51, large)-Net in Base 16 — Upper bound on s
There is no (41, 51, large)-net in base 16, because
- 8 times m-reduction [i] would yield (41, 43, large)-net in base 16, but