Best Known (56−13, 56, s)-Nets in Base 16
(56−13, 56, 21862)-Net over F16 — Constructive and digital
Digital (43, 56, 21862)-net over F16, 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
- digital (37, 50, 21845)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- digital (0, 6, 17)-net over F16, using
(56−13, 56, 43691)-Net in Base 16 — Constructive
(43, 56, 43691)-net in base 16, using
- net defined by OOA [i] based on OOA(1656, 43691, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1656, 262147, S16, 13), using
- discarding parts of the base [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on OA(1656, 262147, S16, 13), using
(56−13, 56, 146728)-Net over F16 — Digital
Digital (43, 56, 146728)-net over F16, using
(56−13, 56, large)-Net in Base 16 — Upper bound on s
There is no (43, 56, large)-net in base 16, because
- 11 times m-reduction [i] would yield (43, 45, large)-net in base 16, but