Best Known (71, 71+53, s)-Nets in Base 16
(71, 71+53, 532)-Net over F16 — Constructive and digital
Digital (71, 124, 532)-net over F16, using
- trace code for nets [i] based on digital (9, 62, 266)-net over F256, using
- net from sequence [i] based on digital (9, 265)-sequence over F256, using
(71, 71+53, 1036)-Net over F16 — Digital
Digital (71, 124, 1036)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16124, 1036, F16, 53) (dual of [1036, 912, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(16124, 1041, F16, 53) (dual of [1041, 917, 54]-code), using
- 12 step Varšamov–Edel lengthening with (ri) = (1, 11 times 0) [i] based on linear OA(16123, 1028, F16, 53) (dual of [1028, 905, 54]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16122, 1026, F16, 53) (dual of [1026, 904, 54]-code), using
- trace code [i] based on linear OA(25661, 513, F256, 53) (dual of [513, 452, 54]-code), using
- extended algebraic-geometric code AGe(F,459P) [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,459P) [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- trace code [i] based on linear OA(25661, 513, F256, 53) (dual of [513, 452, 54]-code), using
- linear OA(16122, 1027, F16, 52) (dual of [1027, 905, 53]-code), using Gilbert–Varšamov bound and bm = 16122 > Vbs−1(k−1) = 647 541201 260657 039266 468286 788610 074332 352670 314324 999940 023904 646285 207970 657811 077217 014724 852149 279455 750443 066309 142089 083545 740416 644972 726016 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16122, 1026, F16, 53) (dual of [1026, 904, 54]-code), using
- construction X with Varšamov bound [i] based on
- 12 step Varšamov–Edel lengthening with (ri) = (1, 11 times 0) [i] based on linear OA(16123, 1028, F16, 53) (dual of [1028, 905, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(16124, 1041, F16, 53) (dual of [1041, 917, 54]-code), using
(71, 71+53, 349621)-Net in Base 16 — Upper bound on s
There is no (71, 124, 349622)-net in base 16, because
- 1 times m-reduction [i] would yield (71, 123, 349622)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 12787 361976 024005 817845 529857 261203 797497 037061 366908 904443 396406 161571 625894 457143 512488 293701 513521 451934 528660 992974 110786 204228 477680 029526 234456 > 16123 [i]