Best Known (47−16, 47, s)-Nets in Base 81
(47−16, 47, 66431)-Net over F81 — Constructive and digital
Digital (31, 47, 66431)-net over F81, using
- net defined by OOA [i] based on linear OOA(8147, 66431, F81, 16, 16) (dual of [(66431, 16), 1062849, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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
- OA 8-folding and stacking [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
(47−16, 47, 265724)-Net over F81 — Digital
Digital (31, 47, 265724)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8147, 265724, F81, 2, 16) (dual of [(265724, 2), 531401, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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
- OOA 2-folding [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
(47−16, 47, large)-Net in Base 81 — Upper bound on s
There is no (31, 47, large)-net in base 81, because
- 14 times m-reduction [i] would yield (31, 33, large)-net in base 81, but