Best Known (56, 56+18, s)-Nets in Base 9
(56, 56+18, 1459)-Net over F9 — Constructive and digital
Digital (56, 74, 1459)-net over F9, using
- 92 times duplication [i] based on digital (54, 72, 1459)-net over F9, using
- net defined by OOA [i] based on linear OOA(972, 1459, F9, 18, 18) (dual of [(1459, 18), 26190, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(972, 13131, F9, 18) (dual of [13131, 13059, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(972, 13131, F9, 18) (dual of [13131, 13059, 19]-code), using
- net defined by OOA [i] based on linear OOA(972, 1459, F9, 18, 18) (dual of [(1459, 18), 26190, 19]-NRT-code), using
(56, 56+18, 13138)-Net over F9 — Digital
Digital (56, 74, 13138)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(974, 13138, F9, 18) (dual of [13138, 13064, 19]-code), using
- trace code [i] based on linear OA(8137, 6569, F81, 18) (dual of [6569, 6532, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- trace code [i] based on linear OA(8137, 6569, F81, 18) (dual of [6569, 6532, 19]-code), using
(56, 56+18, large)-Net in Base 9 — Upper bound on s
There is no (56, 74, large)-net in base 9, because
- 16 times m-reduction [i] would yield (56, 58, large)-net in base 9, but