Best Known (42−17, 42, s)-Nets in Base 16
(42−17, 42, 531)-Net over F16 — Constructive and digital
Digital (25, 42, 531)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (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, 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, 17, 257)-net over F256, using
- digital (0, 8, 17)-net over F16, using
(42−17, 42, 702)-Net over F16 — Digital
Digital (25, 42, 702)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1642, 702, F16, 17) (dual of [702, 660, 18]-code), using
- 56 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0, 1, 40 times 0) [i] based on linear OA(1638, 642, F16, 17) (dual of [642, 604, 18]-code), using
- trace code [i] based on linear OA(25619, 321, F256, 17) (dual of [321, 302, 18]-code), using
- extended algebraic-geometric code AGe(F,303P) [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- trace code [i] based on linear OA(25619, 321, F256, 17) (dual of [321, 302, 18]-code), using
- 56 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0, 1, 40 times 0) [i] based on linear OA(1638, 642, F16, 17) (dual of [642, 604, 18]-code), using
(42−17, 42, 372142)-Net in Base 16 — Upper bound on s
There is no (25, 42, 372143)-net in base 16, because
- 1 times m-reduction [i] would yield (25, 41, 372143)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 23 384389 645340 727618 949660 844685 679335 223097 275711 > 1641 [i]