Best Known (50−23, 50, s)-Nets in Base 81
(50−23, 50, 598)-Net over F81 — Constructive and digital
Digital (27, 50, 598)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 598, F81, 23, 23) (dual of [(598, 23), 13704, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-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([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
(50−23, 50, 3289)-Net over F81 — Digital
Digital (27, 50, 3289)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8150, 3289, F81, 2, 23) (dual of [(3289, 2), 6528, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8150, 6578, F81, 23) (dual of [6578, 6528, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-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([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
- OOA 2-folding [i] based on linear OA(8150, 6578, F81, 23) (dual of [6578, 6528, 24]-code), using
(50−23, 50, large)-Net in Base 81 — Upper bound on s
There is no (27, 50, large)-net in base 81, because
- 21 times m-reduction [i] would yield (27, 29, large)-net in base 81, but