Best Known (36, 36+18, s)-Nets in Base 81
(36, 36+18, 59050)-Net over F81 — Constructive and digital
Digital (36, 54, 59050)-net over F81, using
- net defined by OOA [i] based on linear OOA(8154, 59050, F81, 18, 18) (dual of [(59050, 18), 1062846, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8154, 531450, F81, 18) (dual of [531450, 531396, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8154, 531452, F81, 18) (dual of [531452, 531398, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8154, 531452, F81, 18) (dual of [531452, 531398, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8154, 531450, F81, 18) (dual of [531450, 531396, 19]-code), using
(36, 36+18, 265726)-Net over F81 — Digital
Digital (36, 54, 265726)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8154, 265726, F81, 2, 18) (dual of [(265726, 2), 531398, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8154, 531452, F81, 18) (dual of [531452, 531398, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(8154, 531452, F81, 18) (dual of [531452, 531398, 19]-code), using
(36, 36+18, large)-Net in Base 81 — Upper bound on s
There is no (36, 54, large)-net in base 81, because
- 16 times m-reduction [i] would yield (36, 38, large)-net in base 81, but