Best Known (49−15, 49, s)-Nets in Base 81
(49−15, 49, 75923)-Net over F81 — Constructive and digital
Digital (34, 49, 75923)-net over F81, using
- 811 times duplication [i] based on digital (33, 48, 75923)-net over F81, using
- net defined by OOA [i] based on linear OOA(8148, 75923, F81, 15, 15) (dual of [(75923, 15), 1138797, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8148, 531462, F81, 15) (dual of [531462, 531414, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 531465, F81, 15) (dual of [531465, 531417, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [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(8125, 531442, F81, 9) (dual of [531442, 531417, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-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 C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8148, 531465, F81, 15) (dual of [531465, 531417, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8148, 531462, F81, 15) (dual of [531462, 531414, 16]-code), using
- net defined by OOA [i] based on linear OOA(8148, 75923, F81, 15, 15) (dual of [(75923, 15), 1138797, 16]-NRT-code), using
(49−15, 49, 531468)-Net over F81 — Digital
Digital (34, 49, 531468)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8149, 531468, F81, 15) (dual of [531468, 531419, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(7) [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(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(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
(49−15, 49, large)-Net in Base 81 — Upper bound on s
There is no (34, 49, large)-net in base 81, because
- 13 times m-reduction [i] would yield (34, 36, large)-net in base 81, but