Best Known (45, 56, s)-Nets in Base 16
(45, 56, 209733)-Net over F16 — Constructive and digital
Digital (45, 56, 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 (40, 51, 209716)-net over F16, using
- net defined by OOA [i] based on linear OOA(1651, 209716, F16, 11, 11) (dual of [(209716, 11), 2306825, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1651, 1048581, F16, 11) (dual of [1048581, 1048530, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(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(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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(1651, 1048581, F16, 11) (dual of [1048581, 1048530, 12]-code), using
- net defined by OOA [i] based on linear OOA(1651, 209716, F16, 11, 11) (dual of [(209716, 11), 2306825, 12]-NRT-code), using
- digital (0, 5, 17)-net over F16, using
(45, 56, 419431)-Net in Base 16 — Constructive
(45, 56, 419431)-net in base 16, using
- base change [i] based on digital (21, 32, 419431)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 419431, F128, 11, 11) (dual of [(419431, 11), 4613709, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
- net defined by OOA [i] based on linear OOA(12832, 419431, F128, 11, 11) (dual of [(419431, 11), 4613709, 12]-NRT-code), using
(45, 56, 1670930)-Net over F16 — Digital
Digital (45, 56, 1670930)-net over F16, using
(45, 56, large)-Net in Base 16 — Upper bound on s
There is no (45, 56, large)-net in base 16, because
- 9 times m-reduction [i] would yield (45, 47, large)-net in base 16, but