Best Known (48, 48+13, s)-Nets in Base 81
(48, 48+13, 1400288)-Net over F81 — Constructive and digital
Digital (48, 61, 1400288)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 2188)-net over F81, using
- net defined by OOA [i] based on linear OOA(8112, 2188, F81, 6, 6) (dual of [(2188, 6), 13116, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8112, 6564, F81, 6) (dual of [6564, 6552, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8112, 6566, F81, 6) (dual of [6566, 6554, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [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]
- 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(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(8112, 6566, F81, 6) (dual of [6566, 6554, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(8112, 6564, F81, 6) (dual of [6564, 6552, 7]-code), using
- net defined by OOA [i] based on linear OOA(8112, 2188, F81, 6, 6) (dual of [(2188, 6), 13116, 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 (6, 12, 2188)-net over F81, using
(48, 48+13, large)-Net over F81 — Digital
Digital (48, 61, large)-net over F81, using
- 4 times m-reduction [i] based on digital (48, 65, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
(48, 48+13, large)-Net in Base 81 — Upper bound on s
There is no (48, 61, large)-net in base 81, because
- 11 times m-reduction [i] would yield (48, 50, large)-net in base 81, but