Best Known (21, 32, s)-Nets in Base 16
(21, 32, 1028)-Net over F16 — Constructive and digital
Digital (21, 32, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 5, 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, 5, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
- digital (5, 10, 514)-net over F16, using
(21, 32, 3879)-Net over F16 — Digital
Digital (21, 32, 3879)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1632, 3879, F16, 11) (dual of [3879, 3847, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(1632, 4104, F16, 11) (dual of [4104, 4072, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(1631, 4097, F16, 11) (dual of [4097, 4066, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1625, 4097, F16, 9) (dual of [4097, 4072, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1632, 4104, F16, 11) (dual of [4104, 4072, 12]-code), using
(21, 32, 5073278)-Net in Base 16 — Upper bound on s
There is no (21, 32, 5073279)-net in base 16, because
- 1 times m-reduction [i] would yield (21, 31, 5073279)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 21 267663 528639 086683 133622 851058 455926 > 1631 [i]