Best Known (33, 50, s)-Nets in Base 16
(33, 50, 1028)-Net over F16 — Constructive and digital
Digital (33, 50, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
- digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
- digital (8, 16, 514)-net over F16, using
(33, 50, 3667)-Net over F16 — Digital
Digital (33, 50, 3667)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1650, 3667, F16, 17) (dual of [3667, 3617, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 4104, F16, 17) (dual of [4104, 4054, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- 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(1643, 4097, F16, 15) (dual of [4097, 4054, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1650, 4104, F16, 17) (dual of [4104, 4054, 18]-code), using
(33, 50, 5954336)-Net in Base 16 — Upper bound on s
There is no (33, 50, 5954337)-net in base 16, because
- 1 times m-reduction [i] would yield (33, 49, 5954337)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 100433 636988 547397 274409 549646 512677 897519 578264 246532 854716 > 1649 [i]