Best Known (44−20, 44, s)-Nets in Base 81
(44−20, 44, 657)-Net over F81 — Constructive and digital
Digital (24, 44, 657)-net over F81, using
- t-expansion [i] based on digital (23, 44, 657)-net over F81, using
- net defined by OOA [i] based on linear OOA(8144, 657, F81, 21, 21) (dual of [(657, 21), 13753, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8144, 6571, F81, 21) (dual of [6571, 6527, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 6573, F81, 21) (dual of [6573, 6529, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [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 C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 6573, F81, 21) (dual of [6573, 6529, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8144, 6571, F81, 21) (dual of [6571, 6527, 22]-code), using
- net defined by OOA [i] based on linear OOA(8144, 657, F81, 21, 21) (dual of [(657, 21), 13753, 22]-NRT-code), using
(44−20, 44, 3413)-Net over F81 — Digital
Digital (24, 44, 3413)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8144, 3413, F81, 20) (dual of [3413, 3369, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 6578, F81, 20) (dual of [6578, 6534, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 6578, F81, 20) (dual of [6578, 6534, 21]-code), using
(44−20, 44, large)-Net in Base 81 — Upper bound on s
There is no (24, 44, large)-net in base 81, because
- 18 times m-reduction [i] would yield (24, 26, large)-net in base 81, but