Best Known (82−17, 82, s)-Nets in Base 81
(82−17, 82, 1050217)-Net over F81 — Constructive and digital
Digital (65, 82, 1050217)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (9, 17, 1642)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 1642, F81, 8, 8) (dual of [(1642, 8), 13119, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8117, 6568, F81, 8) (dual of [6568, 6551, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(819, 6561, F81, 5) (dual of [6561, 6552, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8117, 6568, F81, 8) (dual of [6568, 6551, 9]-code), using
- net defined by OOA [i] based on linear OOA(8117, 1642, F81, 8, 8) (dual of [(1642, 8), 13119, 9]-NRT-code), using
- digital (48, 65, 1048575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (9, 17, 1642)-net over F81, using
(82−17, 82, large)-Net over F81 — Digital
Digital (65, 82, large)-net over F81, using
- 811 times duplication [i] based on digital (64, 81, large)-net over F81, using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
(82−17, 82, large)-Net in Base 81 — Upper bound on s
There is no (65, 82, large)-net in base 81, because
- 15 times m-reduction [i] would yield (65, 67, large)-net in base 81, but