Best Known (49, 49+16, s)-Nets in Base 9
(49, 49+16, 1641)-Net over F9 — Constructive and digital
Digital (49, 65, 1641)-net over F9, using
- 91 times duplication [i] based on digital (48, 64, 1641)-net over F9, using
- net defined by OOA [i] based on linear OOA(964, 1641, F9, 16, 16) (dual of [(1641, 16), 26192, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(964, 13128, F9, 16) (dual of [13128, 13064, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(964, 13132, F9, 16) (dual of [13132, 13068, 17]-code), using
- trace code [i] based on linear OA(8132, 6566, F81, 16) (dual of [6566, 6534, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 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(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8132, 6566, F81, 16) (dual of [6566, 6534, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(964, 13132, F9, 16) (dual of [13132, 13068, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(964, 13128, F9, 16) (dual of [13128, 13064, 17]-code), using
- net defined by OOA [i] based on linear OOA(964, 1641, F9, 16, 16) (dual of [(1641, 16), 26192, 17]-NRT-code), using
(49, 49+16, 13134)-Net over F9 — Digital
Digital (49, 65, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(965, 13134, F9, 16) (dual of [13134, 13069, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(964, 13132, F9, 16) (dual of [13132, 13068, 17]-code), using
- trace code [i] based on linear OA(8132, 6566, F81, 16) (dual of [6566, 6534, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 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(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8132, 6566, F81, 16) (dual of [6566, 6534, 17]-code), using
- linear OA(964, 13133, F9, 15) (dual of [13133, 13069, 16]-code), using Gilbert–Varšamov bound and bm = 964 > Vbs−1(k−1) = 227264 239100 447614 177158 934900 648413 269572 351113 717315 807969 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(964, 13132, F9, 16) (dual of [13132, 13068, 17]-code), using
- construction X with Varšamov bound [i] based on
(49, 49+16, large)-Net in Base 9 — Upper bound on s
There is no (49, 65, large)-net in base 9, because
- 14 times m-reduction [i] would yield (49, 51, large)-net in base 9, but