Best Known (54, 54+13, s)-Nets in Base 81
(54, 54+13, 1575250)-Net over F81 — Constructive and digital
Digital (54, 67, 1575250)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (12, 18, 177150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8118, 177150, F81, 6, 6) (dual of [(177150, 6), 1062882, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8118, 531450, F81, 6) (dual of [531450, 531432, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8118, 531452, F81, 6) (dual of [531452, 531434, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(817, 531441, F81, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(812, 11, F81, 2) (dual of [11, 9, 3]-code or 11-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(8118, 531452, F81, 6) (dual of [531452, 531434, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(8118, 531450, F81, 6) (dual of [531450, 531432, 7]-code), using
- net defined by OOA [i] based on linear OOA(8118, 177150, F81, 6, 6) (dual of [(177150, 6), 1062882, 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 (12, 18, 177150)-net over F81, using
(54, 54+13, large)-Net over F81 — Digital
Digital (54, 67, large)-net over F81, using
- 6 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
(54, 54+13, large)-Net in Base 81 — Upper bound on s
There is no (54, 67, large)-net in base 81, because
- 11 times m-reduction [i] would yield (54, 56, large)-net in base 81, but