Best Known (59−18, 59, s)-Nets in Base 81
(59−18, 59, 59052)-Net over F81 — Constructive and digital
Digital (41, 59, 59052)-net over F81, using
- 811 times duplication [i] based on digital (40, 58, 59052)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 59052, F81, 18, 18) (dual of [(59052, 18), 1062878, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8158, 531468, F81, 18) (dual of [531468, 531410, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(17) ⊂ Ce(10) [i] based on
- OA 9-folding and stacking [i] based on linear OA(8158, 531468, F81, 18) (dual of [531468, 531410, 19]-code), using
- net defined by OOA [i] based on linear OOA(8158, 59052, F81, 18, 18) (dual of [(59052, 18), 1062878, 19]-NRT-code), using
(59−18, 59, 531472)-Net over F81 — Digital
Digital (41, 59, 531472)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 531472, F81, 18) (dual of [531472, 531413, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(9) [i] based on
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(817, 31, F81, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to Ce(17) ⊂ Ce(9) [i] based on
(59−18, 59, large)-Net in Base 81 — Upper bound on s
There is no (41, 59, large)-net in base 81, because
- 16 times m-reduction [i] would yield (41, 43, large)-net in base 81, but