Best Known (56, 56+39, s)-Nets in Base 16
(56, 56+39, 530)-Net over F16 — Constructive and digital
Digital (56, 95, 530)-net over F16, using
- 1 times m-reduction [i] based on digital (56, 96, 530)-net over F16, using
- trace code for nets [i] based on digital (8, 48, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
- trace code for nets [i] based on digital (8, 48, 265)-net over F256, using
(56, 56+39, 1055)-Net over F16 — Digital
Digital (56, 95, 1055)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1695, 1055, F16, 39) (dual of [1055, 960, 40]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (1, 27 times 0) [i] based on linear OA(1694, 1026, F16, 39) (dual of [1026, 932, 40]-code), using
- trace code [i] based on linear OA(25647, 513, F256, 39) (dual of [513, 466, 40]-code), using
- extended algebraic-geometric code AGe(F,473P) [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,473P) [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- trace code [i] based on linear OA(25647, 513, F256, 39) (dual of [513, 466, 40]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (1, 27 times 0) [i] based on linear OA(1694, 1026, F16, 39) (dual of [1026, 932, 40]-code), using
(56, 56+39, 479005)-Net in Base 16 — Upper bound on s
There is no (56, 95, 479006)-net in base 16, because
- 1 times m-reduction [i] would yield (56, 94, 479006)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 153919 722309 167726 863372 797498 764447 801510 097702 891516 294702 710071 579304 051709 364302 122206 667681 466905 647047 875686 > 1694 [i]