Best Known (22−10, 22, s)-Nets in Base 81
(22−10, 22, 1314)-Net over F81 — Constructive and digital
Digital (12, 22, 1314)-net over F81, using
- net defined by OOA [i] based on linear OOA(8122, 1314, F81, 10, 10) (dual of [(1314, 10), 13118, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8122, 6570, F81, 10) (dual of [6570, 6548, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 6572, F81, 10) (dual of [6572, 6550, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8122, 6572, F81, 10) (dual of [6572, 6550, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8122, 6570, F81, 10) (dual of [6570, 6548, 11]-code), using
(22−10, 22, 4809)-Net over F81 — Digital
Digital (12, 22, 4809)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8122, 4809, F81, 10) (dual of [4809, 4787, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 6572, F81, 10) (dual of [6572, 6550, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8122, 6572, F81, 10) (dual of [6572, 6550, 11]-code), using
(22−10, 22, 8129806)-Net in Base 81 — Upper bound on s
There is no (12, 22, 8129807)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 969773 735895 027015 798059 765975 863901 394801 > 8122 [i]