Best Known (57, 72, s)-Nets in Base 81
(57, 72, 1200560)-Net over F81 — Constructive and digital
Digital (57, 72, 1200560)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (8, 15, 2189)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 2189, F81, 7, 7) (dual of [(2189, 7), 15308, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8115, 6568, F81, 7) (dual of [6568, 6553, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6569, F81, 7) (dual of [6569, 6554, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(8115, 6569, F81, 7) (dual of [6569, 6554, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8115, 6568, F81, 7) (dual of [6568, 6553, 8]-code), using
- net defined by OOA [i] based on linear OOA(8115, 2189, F81, 7, 7) (dual of [(2189, 7), 15308, 8]-NRT-code), using
- digital (42, 57, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (8, 15, 2189)-net over F81, using
(57, 72, large)-Net over F81 — Digital
Digital (57, 72, large)-net over F81, using
- 5 times m-reduction [i] based on digital (57, 77, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
(57, 72, large)-Net in Base 81 — Upper bound on s
There is no (57, 72, large)-net in base 81, because
- 13 times m-reduction [i] would yield (57, 59, large)-net in base 81, but