Best Known (68−29, 68, s)-Nets in Base 16
(68−29, 68, 524)-Net over F16 — Constructive and digital
Digital (39, 68, 524)-net over F16, using
- trace code for nets [i] based on digital (5, 34, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(68−29, 68, 682)-Net over F16 — Digital
Digital (39, 68, 682)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 682, F16, 29) (dual of [682, 614, 30]-code), using
- 34 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 8 times 0, 1, 20 times 0) [i] based on linear OA(1662, 642, F16, 29) (dual of [642, 580, 30]-code), using
- trace code [i] based on linear OA(25631, 321, F256, 29) (dual of [321, 290, 30]-code), using
- extended algebraic-geometric code AGe(F,291P) [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- trace code [i] based on linear OA(25631, 321, F256, 29) (dual of [321, 290, 30]-code), using
- 34 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 8 times 0, 1, 20 times 0) [i] based on linear OA(1662, 642, F16, 29) (dual of [642, 580, 30]-code), using
(68−29, 68, 233306)-Net in Base 16 — Upper bound on s
There is no (39, 68, 233307)-net in base 16, because
- 1 times m-reduction [i] would yield (39, 67, 233307)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 474 298506 713321 772417 507434 345176 070565 517484 759428 115981 159776 746030 569094 877396 > 1667 [i]