Best Known (26, 26+12, s)-Nets in Base 81
(26, 26+12, 88576)-Net over F81 — Constructive and digital
Digital (26, 38, 88576)-net over F81, using
- 811 times duplication [i] based on digital (25, 37, 88576)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 88576, F81, 12, 12) (dual of [(88576, 12), 1062875, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-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
- OA 6-folding and stacking [i] based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
- net defined by OOA [i] based on linear OOA(8137, 88576, F81, 12, 12) (dual of [(88576, 12), 1062875, 13]-NRT-code), using
(26, 26+12, 531460)-Net over F81 — Digital
Digital (26, 38, 531460)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8138, 531460, F81, 12) (dual of [531460, 531422, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(814, 19, F81, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
(26, 26+12, large)-Net in Base 81 — Upper bound on s
There is no (26, 38, large)-net in base 81, because
- 10 times m-reduction [i] would yield (26, 28, large)-net in base 81, but