Best Known (40, 67, s)-Nets in Base 16
(40, 67, 531)-Net over F16 — Constructive and digital
Digital (40, 67, 531)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 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 (27, 54, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- digital (0, 13, 17)-net over F16, using
(40, 67, 905)-Net over F16 — Digital
Digital (40, 67, 905)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1667, 905, F16, 27) (dual of [905, 838, 28]-code), using
- 316 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 42 times 0, 1, 61 times 0, 1, 76 times 0, 1, 88 times 0) [i] based on linear OA(1656, 578, F16, 27) (dual of [578, 522, 28]-code), using
- trace code [i] based on linear OA(25628, 289, F256, 27) (dual of [289, 261, 28]-code), using
- extended algebraic-geometric code AGe(F,261P) [i] based on function field F/F256 with g(F) = 1 and N(F) ≥ 289, using
- trace code [i] based on linear OA(25628, 289, F256, 27) (dual of [289, 261, 28]-code), using
- 316 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 42 times 0, 1, 61 times 0, 1, 76 times 0, 1, 88 times 0) [i] based on linear OA(1656, 578, F16, 27) (dual of [578, 522, 28]-code), using
(40, 67, 490380)-Net in Base 16 — Upper bound on s
There is no (40, 67, 490381)-net in base 16, because
- 1 times m-reduction [i] would yield (40, 66, 490381)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 29 643014 163086 720536 893931 794318 406541 002967 086822 419864 188662 494417 261968 278096 > 1666 [i]