Best Known (51−16, 51, s)-Nets in Base 81
(51−16, 51, 66433)-Net over F81 — Constructive and digital
Digital (35, 51, 66433)-net over F81, using
- net defined by OOA [i] based on linear OOA(8151, 66433, F81, 16, 16) (dual of [(66433, 16), 1062877, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8151, 531464, F81, 16) (dual of [531464, 531413, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [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(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- OA 8-folding and stacking [i] based on linear OA(8151, 531464, F81, 16) (dual of [531464, 531413, 17]-code), using
(51−16, 51, 494744)-Net over F81 — Digital
Digital (35, 51, 494744)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8151, 494744, F81, 16) (dual of [494744, 494693, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8151, 531464, F81, 16) (dual of [531464, 531413, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [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(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8151, 531464, F81, 16) (dual of [531464, 531413, 17]-code), using
(51−16, 51, large)-Net in Base 81 — Upper bound on s
There is no (35, 51, large)-net in base 81, because
- 14 times m-reduction [i] would yield (35, 37, large)-net in base 81, but