Best Known (49, 49+13, s)-Nets in Base 81
(49, 49+13, 1400289)-Net over F81 — Constructive and digital
Digital (49, 62, 1400289)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 2189)-net over F81, using
- net defined by OOA [i] based on linear OOA(8113, 2189, F81, 6, 6) (dual of [(2189, 6), 13121, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8113, 6567, F81, 6) (dual of [6567, 6554, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 6569, F81, 6) (dual of [6569, 6556, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [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(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(8113, 6569, F81, 6) (dual of [6569, 6556, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(8113, 6567, F81, 6) (dual of [6567, 6554, 7]-code), using
- net defined by OOA [i] based on linear OOA(8113, 2189, F81, 6, 6) (dual of [(2189, 6), 13121, 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 (7, 13, 2189)-net over F81, using
(49, 49+13, large)-Net over F81 — Digital
Digital (49, 62, large)-net over F81, using
- t-expansion [i] based on digital (48, 62, large)-net over F81, using
- 3 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
- 3 times m-reduction [i] based on digital (48, 65, large)-net over F81, using
(49, 49+13, large)-Net in Base 81 — Upper bound on s
There is no (49, 62, large)-net in base 81, because
- 11 times m-reduction [i] would yield (49, 51, large)-net in base 81, but