Best Known (77, 77+20, s)-Nets in Base 16
(77, 77+20, 104860)-Net over F16 — Constructive and digital
Digital (77, 97, 104860)-net over F16, using
- 161 times duplication [i] based on digital (76, 96, 104860)-net over F16, using
- net defined by OOA [i] based on linear OOA(1696, 104860, F16, 20, 20) (dual of [(104860, 20), 2097104, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1696, 1048600, F16, 20) (dual of [1048600, 1048504, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(165, 24, F16, 4) (dual of [24, 19, 5]-code), using
- extended algebraic-geometric code AGe(F,19P) [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- OA 10-folding and stacking [i] based on linear OA(1696, 1048600, F16, 20) (dual of [1048600, 1048504, 21]-code), using
- net defined by OOA [i] based on linear OOA(1696, 104860, F16, 20, 20) (dual of [(104860, 20), 2097104, 21]-NRT-code), using
(77, 77+20, 1048603)-Net over F16 — Digital
Digital (77, 97, 1048603)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1697, 1048603, F16, 20) (dual of [1048603, 1048506, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1696, 1048601, F16, 20) (dual of [1048601, 1048505, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(165, 25, F16, 4) (dual of [25, 20, 5]-code), using
- extended algebraic-geometric code AGe(F,20P) [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 25, using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(1696, 1048602, F16, 19) (dual of [1048602, 1048506, 20]-code), using Gilbert–Varšamov bound and bm = 1696 > Vbs−1(k−1) = 542280 000575 358323 942605 879162 889789 734840 954287 444859 359296 245919 982911 586701 365133 872837 984030 545433 412317 546516 [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(1696, 1048601, F16, 20) (dual of [1048601, 1048505, 21]-code), using
- construction X with Varšamov bound [i] based on
(77, 77+20, large)-Net in Base 16 — Upper bound on s
There is no (77, 97, large)-net in base 16, because
- 18 times m-reduction [i] would yield (77, 79, large)-net in base 16, but