Best Known (52−16, 52, s)-Nets in Base 81
(52−16, 52, 66433)-Net over F81 — Constructive and digital
Digital (36, 52, 66433)-net over F81, using
- 811 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(8151, 66433, F81, 16, 16) (dual of [(66433, 16), 1062877, 17]-NRT-code), using
(52−16, 52, 531468)-Net over F81 — Digital
Digital (36, 52, 531468)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8152, 531468, F81, 16) (dual of [531468, 531416, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [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(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
(52−16, 52, large)-Net in Base 81 — Upper bound on s
There is no (36, 52, large)-net in base 81, because
- 14 times m-reduction [i] would yield (36, 38, large)-net in base 81, but