Best Known (56−19, 56, s)-Nets in Base 16
(56−19, 56, 1028)-Net over F16 — Constructive and digital
Digital (37, 56, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- digital (19, 38, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- digital (9, 18, 514)-net over F16, using
(56−19, 56, 3754)-Net over F16 — Digital
Digital (37, 56, 3754)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1656, 3754, F16, 19) (dual of [3754, 3698, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1656, 4104, F16, 19) (dual of [4104, 4048, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(1655, 4097, F16, 19) (dual of [4097, 4042, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(1649, 4097, F16, 17) (dual of [4097, 4048, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1656, 4104, F16, 19) (dual of [4104, 4048, 20]-code), using
(56−19, 56, 6312056)-Net in Base 16 — Upper bound on s
There is no (37, 56, 6312057)-net in base 16, because
- 1 times m-reduction [i] would yield (37, 55, 6312057)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 1 684997 867241 934169 530664 134520 725439 266620 062565 570109 172885 718496 > 1655 [i]