Best Known (81−17, 81, s)-Nets in Base 16
(81−17, 81, 131074)-Net over F16 — Constructive and digital
Digital (64, 81, 131074)-net over F16, using
- 162 times duplication [i] based on digital (62, 79, 131074)-net over F16, using
- net defined by OOA [i] based on linear OOA(1679, 131074, F16, 17, 17) (dual of [(131074, 17), 2228179, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1679, 1048593, F16, 17) (dual of [1048593, 1048514, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1679, 1048594, F16, 17) (dual of [1048594, 1048515, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(1679, 1048594, F16, 17) (dual of [1048594, 1048515, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1679, 1048593, F16, 17) (dual of [1048593, 1048514, 18]-code), using
- net defined by OOA [i] based on linear OOA(1679, 131074, F16, 17, 17) (dual of [(131074, 17), 2228179, 18]-NRT-code), using
(81−17, 81, 1048601)-Net over F16 — Digital
Digital (64, 81, 1048601)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1681, 1048601, F16, 17) (dual of [1048601, 1048520, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
(81−17, 81, large)-Net in Base 16 — Upper bound on s
There is no (64, 81, large)-net in base 16, because
- 15 times m-reduction [i] would yield (64, 66, large)-net in base 16, but