Best Known (50−21, 50, s)-Nets in Base 81
(50−21, 50, 659)-Net over F81 — Constructive and digital
Digital (29, 50, 659)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 659, F81, 21, 21) (dual of [(659, 21), 13789, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8150, 6591, F81, 21) (dual of [6591, 6541, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- 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(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-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(8150, 6591, F81, 21) (dual of [6591, 6541, 22]-code), using
(50−21, 50, 6591)-Net over F81 — Digital
Digital (29, 50, 6591)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8150, 6591, F81, 21) (dual of [6591, 6541, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- 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(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-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
(50−21, 50, large)-Net in Base 81 — Upper bound on s
There is no (29, 50, large)-net in base 81, because
- 19 times m-reduction [i] would yield (29, 31, large)-net in base 81, but