Best Known (46−16, 46, s)-Nets in Base 81
(46−16, 46, 66430)-Net over F81 — Constructive and digital
Digital (30, 46, 66430)-net over F81, using
- net defined by OOA [i] based on linear OOA(8146, 66430, F81, 16, 16) (dual of [(66430, 16), 1062834, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8146, 531440, F81, 16) (dual of [531440, 531394, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(8146, 531440, F81, 16) (dual of [531440, 531394, 17]-code), using
(46−16, 46, 204076)-Net over F81 — Digital
Digital (30, 46, 204076)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8146, 204076, F81, 2, 16) (dual of [(204076, 2), 408106, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 265722, F81, 2, 16) (dual of [(265722, 2), 531398, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8146, 531444, F81, 16) (dual of [531444, 531398, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(8146, 531444, F81, 16) (dual of [531444, 531398, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 265722, F81, 2, 16) (dual of [(265722, 2), 531398, 17]-NRT-code), using
(46−16, 46, large)-Net in Base 81 — Upper bound on s
There is no (30, 46, large)-net in base 81, because
- 14 times m-reduction [i] would yield (30, 32, large)-net in base 81, but