Best Known (60−13, 60, s)-Nets in Base 81
(60−13, 60, 1400287)-Net over F81 — Constructive and digital
Digital (47, 60, 1400287)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 2187)-net over F81, using
- net defined by OOA [i] based on linear OOA(8111, 2187, F81, 6, 6) (dual of [(2187, 6), 13111, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OA 3-folding and stacking [i] based on linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using
- net defined by OOA [i] based on linear OOA(8111, 2187, F81, 6, 6) (dual of [(2187, 6), 13111, 7]-NRT-code), using
- digital (36, 49, 1398100)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (5, 11, 2187)-net over F81, using
(60−13, 60, large)-Net over F81 — Digital
Digital (47, 60, large)-net over F81, using
- t-expansion [i] based on digital (45, 60, large)-net over F81, using
- 1 times m-reduction [i] based on digital (45, 61, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- 1 times m-reduction [i] based on digital (45, 61, large)-net over F81, using
(60−13, 60, large)-Net in Base 81 — Upper bound on s
There is no (47, 60, large)-net in base 81, because
- 11 times m-reduction [i] would yield (47, 49, large)-net in base 81, but