Best Known (70−21, 70, s)-Nets in Base 81
(70−21, 70, 53148)-Net over F81 — Constructive and digital
Digital (49, 70, 53148)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 53148, F81, 21, 21) (dual of [(53148, 21), 1116038, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8170, 531481, F81, 21) (dual of [531481, 531411, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- 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(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(819, 39, F81, 9) (dual of [39, 30, 10]-code or 39-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(8170, 531481, F81, 21) (dual of [531481, 531411, 22]-code), using
(70−21, 70, 531481)-Net over F81 — Digital
Digital (49, 70, 531481)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 531481, F81, 21) (dual of [531481, 531411, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- 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(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(819, 39, F81, 9) (dual of [39, 30, 10]-code or 39-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
(70−21, 70, large)-Net in Base 81 — Upper bound on s
There is no (49, 70, large)-net in base 81, because
- 19 times m-reduction [i] would yield (49, 51, large)-net in base 81, but