Best Known (80−17, 80, s)-Nets in Base 16
(80−17, 80, 131074)-Net over F16 — Constructive and digital
Digital (63, 80, 131074)-net over F16, using
- 161 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
(80−17, 80, 940525)-Net over F16 — Digital
Digital (63, 80, 940525)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1680, 940525, F16, 17) (dual of [940525, 940445, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 1048595, F16, 17) (dual of [1048595, 1048515, 18]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(1679, 1048594, F16, 17) (dual of [1048594, 1048515, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 1048595, F16, 17) (dual of [1048595, 1048515, 18]-code), using
(80−17, 80, large)-Net in Base 16 — Upper bound on s
There is no (63, 80, large)-net in base 16, because
- 15 times m-reduction [i] would yield (63, 65, large)-net in base 16, but