Best Known (72, 72+54, s)-Nets in Base 16
(72, 72+54, 532)-Net over F16 — Constructive and digital
Digital (72, 126, 532)-net over F16, using
- trace code for nets [i] based on digital (9, 63, 266)-net over F256, using
- net from sequence [i] based on digital (9, 265)-sequence over F256, using
(72, 72+54, 1032)-Net over F16 — Digital
Digital (72, 126, 1032)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16126, 1032, F16, 54) (dual of [1032, 906, 55]-code), using
- discarding factors / shortening the dual code based on linear OA(16126, 1041, F16, 54) (dual of [1041, 915, 55]-code), using
- 12 step Varšamov–Edel lengthening with (ri) = (1, 11 times 0) [i] based on linear OA(16125, 1028, F16, 54) (dual of [1028, 903, 55]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16124, 1026, F16, 54) (dual of [1026, 902, 55]-code), using
- trace code [i] based on linear OA(25662, 513, F256, 54) (dual of [513, 451, 55]-code), using
- extended algebraic-geometric code AGe(F,458P) [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,458P) [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- trace code [i] based on linear OA(25662, 513, F256, 54) (dual of [513, 451, 55]-code), using
- linear OA(16124, 1027, F16, 53) (dual of [1027, 903, 54]-code), using Gilbert–Varšamov bound and bm = 16124 > Vbs−1(k−1) = 182134 111981 420724 348257 878194 904826 751861 179506 358632 144141 150739 989482 862846 490472 407677 553358 237271 715754 039460 097802 273160 412662 561705 707472 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(16124, 1026, F16, 54) (dual of [1026, 902, 55]-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(16125, 1028, F16, 54) (dual of [1028, 903, 55]-code), using
- discarding factors / shortening the dual code based on linear OA(16126, 1041, F16, 54) (dual of [1041, 915, 55]-code), using
(72, 72+54, 303054)-Net in Base 16 — Upper bound on s
There is no (72, 126, 303055)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 52 376090 778627 957749 734854 363449 400323 094000 082027 952003 534568 477655 312255 399792 603258 031109 281329 253361 986360 077556 470883 793524 330155 676054 260913 633776 > 16126 [i]