Best Known (26−12, 26, s)-Nets in Base 81
(26−12, 26, 1095)-Net over F81 — Constructive and digital
Digital (14, 26, 1095)-net over F81, using
- net defined by OOA [i] based on linear OOA(8126, 1095, F81, 12, 12) (dual of [(1095, 12), 13114, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8126, 6570, F81, 12) (dual of [6570, 6544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8126, 6572, F81, 12) (dual of [6572, 6546, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8126, 6572, F81, 12) (dual of [6572, 6546, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8126, 6570, F81, 12) (dual of [6570, 6544, 13]-code), using
(26−12, 26, 3339)-Net over F81 — Digital
Digital (14, 26, 3339)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8126, 3339, F81, 12) (dual of [3339, 3313, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8126, 6572, F81, 12) (dual of [6572, 6546, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8126, 6572, F81, 12) (dual of [6572, 6546, 13]-code), using
(26−12, 26, 6970014)-Net in Base 81 — Upper bound on s
There is no (14, 26, 6970015)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 41 745595 880682 369485 979871 281275 583799 735766 727201 > 8126 [i]