Best Known (41−18, 41, s)-Nets in Base 81
(41−18, 41, 731)-Net over F81 — Constructive and digital
Digital (23, 41, 731)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 731, F81, 18, 18) (dual of [(731, 18), 13117, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8141, 6579, F81, 18) (dual of [6579, 6538, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, 6581, F81, 18) (dual of [6581, 6540, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-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
- discarding factors / shortening the dual code based on linear OA(8141, 6581, F81, 18) (dual of [6581, 6540, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8141, 6579, F81, 18) (dual of [6579, 6538, 19]-code), using
(41−18, 41, 5012)-Net over F81 — Digital
Digital (23, 41, 5012)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8141, 5012, F81, 18) (dual of [5012, 4971, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, 6581, F81, 18) (dual of [6581, 6540, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-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
- discarding factors / shortening the dual code based on linear OA(8141, 6581, F81, 18) (dual of [6581, 6540, 19]-code), using
(41−18, 41, large)-Net in Base 81 — Upper bound on s
There is no (23, 41, large)-net in base 81, because
- 16 times m-reduction [i] would yield (23, 25, large)-net in base 81, but