Best Known (68, 117, s)-Nets in Base 16
(68, 117, 532)-Net over F16 — Constructive and digital
Digital (68, 117, 532)-net over F16, using
- 1 times m-reduction [i] based on digital (68, 118, 532)-net over F16, using
- trace code for nets [i] based on digital (9, 59, 266)-net over F256, using
- net from sequence [i] based on digital (9, 265)-sequence over F256, using
- trace code for nets [i] based on digital (9, 59, 266)-net over F256, using
(68, 117, 1101)-Net over F16 — Digital
Digital (68, 117, 1101)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16117, 1101, F16, 49) (dual of [1101, 984, 50]-code), using
- 72 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 18 times 0, 1, 50 times 0) [i] based on linear OA(16114, 1026, F16, 49) (dual of [1026, 912, 50]-code), using
- trace code [i] based on linear OA(25657, 513, F256, 49) (dual of [513, 456, 50]-code), using
- extended algebraic-geometric code AGe(F,463P) [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- extended algebraic-geometric code AGe(F,463P) [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- trace code [i] based on linear OA(25657, 513, F256, 49) (dual of [513, 456, 50]-code), using
- 72 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 18 times 0, 1, 50 times 0) [i] based on linear OA(16114, 1026, F16, 49) (dual of [1026, 912, 50]-code), using
(68, 117, 431689)-Net in Base 16 — Upper bound on s
There is no (68, 117, 431690)-net in base 16, because
- 1 times m-reduction [i] would yield (68, 116, 431690)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 47 635262 122879 602841 897083 296801 610632 203150 729707 202185 600316 501019 248816 564121 858238 596572 439265 475964 310047 219722 681848 882464 918917 102276 > 16116 [i]