Best Known (63−27, 63, s)-Nets in Base 16
(63−27, 63, 522)-Net over F16 — Constructive and digital
Digital (36, 63, 522)-net over F16, using
- 1 times m-reduction [i] based on digital (36, 64, 522)-net over F16, using
- trace code for nets [i] based on digital (4, 32, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 32, 261)-net over F256, using
(63−27, 63, 645)-Net over F16 — Digital
Digital (36, 63, 645)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1663, 645, F16, 27) (dual of [645, 582, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(1663, 652, F16, 27) (dual of [652, 589, 28]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1662, 650, F16, 27) (dual of [650, 588, 28]-code), using
- trace code [i] based on linear OA(25631, 325, F256, 27) (dual of [325, 294, 28]-code), using
- construction X applied to AG(F,292P) ⊂ AG(F,295P) [i] based on
- linear OA(25629, 320, F256, 27) (dual of [320, 291, 28]-code), using algebraic-geometric code AG(F,292P) [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- linear OA(25626, 320, F256, 24) (dual of [320, 294, 25]-code), using algebraic-geometric code AG(F,295P) [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321 (see above)
- linear OA(2562, 5, F256, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to AG(F,292P) ⊂ AG(F,295P) [i] based on
- trace code [i] based on linear OA(25631, 325, F256, 27) (dual of [325, 294, 28]-code), using
- linear OA(1662, 651, F16, 26) (dual of [651, 589, 27]-code), using Gilbert–Varšamov bound and bm = 1662 > Vbs−1(k−1) = 215 088332 340518 373670 892898 496421 190733 399248 366357 064330 490104 198789 980376 [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(1662, 650, F16, 27) (dual of [650, 588, 28]-code), using
- construction X with Varšamov bound [i] based on
- discarding factors / shortening the dual code based on linear OA(1663, 652, F16, 27) (dual of [652, 589, 28]-code), using
(63−27, 63, 208942)-Net in Base 16 — Upper bound on s
There is no (36, 63, 208943)-net in base 16, because
- 1 times m-reduction [i] would yield (36, 62, 208943)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 452 327850 033595 092447 132960 977612 846134 149093 182348 531616 516491 264930 626186 > 1662 [i]