Best Known (59−27, 59, s)-Nets in Base 81
(59−27, 59, 506)-Net over F81 — Constructive and digital
Digital (32, 59, 506)-net over F81, using
- 811 times duplication [i] based on digital (31, 58, 506)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 506, F81, 27, 27) (dual of [(506, 27), 13604, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(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([0,13]) ⊂ C([0,10]) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- net defined by OOA [i] based on linear OOA(8158, 506, F81, 27, 27) (dual of [(506, 27), 13604, 28]-NRT-code), using
(59−27, 59, 3394)-Net over F81 — Digital
Digital (32, 59, 3394)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 3394, F81, 27) (dual of [3394, 3335, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 6581, F81, 27) (dual of [6581, 6522, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(26) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(8159, 6581, F81, 27) (dual of [6581, 6522, 28]-code), using
(59−27, 59, large)-Net in Base 81 — Upper bound on s
There is no (32, 59, large)-net in base 81, because
- 25 times m-reduction [i] would yield (32, 34, large)-net in base 81, but