Best Known (48−16, 48, s)-Nets in Base 81
(48−16, 48, 66431)-Net over F81 — Constructive and digital
Digital (32, 48, 66431)-net over F81, using
- 811 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(8147, 66431, F81, 16, 16) (dual of [(66431, 16), 1062849, 17]-NRT-code), using
(48−16, 48, 265726)-Net over F81 — Digital
Digital (32, 48, 265726)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8148, 265726, F81, 2, 16) (dual of [(265726, 2), 531404, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8148, 531452, F81, 16) (dual of [531452, 531404, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(15) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(8148, 531452, F81, 16) (dual of [531452, 531404, 17]-code), using
(48−16, 48, large)-Net in Base 81 — Upper bound on s
There is no (32, 48, large)-net in base 81, because
- 14 times m-reduction [i] would yield (32, 34, large)-net in base 81, but