Best Known (47−15, 47, s)-Nets in Base 81
(47−15, 47, 75922)-Net over F81 — Constructive and digital
Digital (32, 47, 75922)-net over F81, using
- 811 times duplication [i] based on digital (31, 46, 75922)-net over F81, using
- net defined by OOA [i] based on linear OOA(8146, 75922, F81, 15, 15) (dual of [(75922, 15), 1138784, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8146, 531455, F81, 15) (dual of [531455, 531409, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8146, 531457, F81, 15) (dual of [531457, 531411, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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 C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8146, 531457, F81, 15) (dual of [531457, 531411, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8146, 531455, F81, 15) (dual of [531455, 531409, 16]-code), using
- net defined by OOA [i] based on linear OOA(8146, 75922, F81, 15, 15) (dual of [(75922, 15), 1138784, 16]-NRT-code), using
(47−15, 47, 401246)-Net over F81 — Digital
Digital (32, 47, 401246)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8147, 401246, F81, 15) (dual of [401246, 401199, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 531460, F81, 15) (dual of [531460, 531413, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- 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(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8147, 531460, F81, 15) (dual of [531460, 531413, 16]-code), using
(47−15, 47, large)-Net in Base 81 — Upper bound on s
There is no (32, 47, large)-net in base 81, because
- 13 times m-reduction [i] would yield (32, 34, large)-net in base 81, but