Best Known (82, 82+25, s)-Nets in Base 9
(82, 82+25, 1095)-Net over F9 — Constructive and digital
Digital (82, 107, 1095)-net over F9, using
- 93 times duplication [i] based on digital (79, 104, 1095)-net over F9, using
- net defined by OOA [i] based on linear OOA(9104, 1095, F9, 25, 25) (dual of [(1095, 25), 27271, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9104, 13141, F9, 25) (dual of [13141, 13037, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 13146, F9, 25) (dual of [13146, 13042, 26]-code), using
- trace code [i] based on linear OA(8152, 6573, F81, 25) (dual of [6573, 6521, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(8152, 6573, F81, 25) (dual of [6573, 6521, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 13146, F9, 25) (dual of [13146, 13042, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9104, 13141, F9, 25) (dual of [13141, 13037, 26]-code), using
- net defined by OOA [i] based on linear OOA(9104, 1095, F9, 25, 25) (dual of [(1095, 25), 27271, 26]-NRT-code), using
(82, 82+25, 22021)-Net over F9 — Digital
Digital (82, 107, 22021)-net over F9, using
(82, 82+25, large)-Net in Base 9 — Upper bound on s
There is no (82, 107, large)-net in base 9, because
- 23 times m-reduction [i] would yield (82, 84, large)-net in base 9, but