Best Known (49−22, 49, s)-Nets in Base 81
(49−22, 49, 598)-Net over F81 — Constructive and digital
Digital (27, 49, 598)-net over F81, using
- 1 times m-reduction [i] based on 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
- net defined by OOA [i] based on linear OOA(8150, 598, F81, 23, 23) (dual of [(598, 23), 13704, 24]-NRT-code), using
(49−22, 49, 3941)-Net over F81 — Digital
Digital (27, 49, 3941)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8149, 3941, F81, 22) (dual of [3941, 3892, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 6581, F81, 22) (dual of [6581, 6532, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8149, 6581, F81, 22) (dual of [6581, 6532, 23]-code), using
(49−22, 49, large)-Net in Base 81 — Upper bound on s
There is no (27, 49, large)-net in base 81, because
- 20 times m-reduction [i] would yield (27, 29, large)-net in base 81, but