Best Known (78−25, 78, s)-Nets in Base 81
(78−25, 78, 44288)-Net over F81 — Constructive and digital
Digital (53, 78, 44288)-net over F81, using
- 812 times duplication [i] based on digital (51, 76, 44288)-net over F81, using
- net defined by OOA [i] based on linear OOA(8176, 44288, F81, 25, 25) (dual of [(44288, 25), 1107124, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8176, 531457, F81, 25) (dual of [531457, 531381, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(8176, 531457, F81, 25) (dual of [531457, 531381, 26]-code), using
- net defined by OOA [i] based on linear OOA(8176, 44288, F81, 25, 25) (dual of [(44288, 25), 1107124, 26]-NRT-code), using
(78−25, 78, 288819)-Net over F81 — Digital
Digital (53, 78, 288819)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8178, 288819, F81, 25) (dual of [288819, 288741, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8178, 531465, F81, 25) (dual of [531465, 531387, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8178, 531465, F81, 25) (dual of [531465, 531387, 26]-code), using
(78−25, 78, large)-Net in Base 81 — Upper bound on s
There is no (53, 78, large)-net in base 81, because
- 23 times m-reduction [i] would yield (53, 55, large)-net in base 81, but