Best Known (33, 57, s)-Nets in Base 81
(33, 57, 549)-Net over F81 — Constructive and digital
Digital (33, 57, 549)-net over F81, using
- 1 times m-reduction [i] based on digital (33, 58, 549)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 549, F81, 25, 25) (dual of [(549, 25), 13667, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8158, 6589, F81, 25) (dual of [6589, 6531, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8158, 6591, F81, 25) (dual of [6591, 6533, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,7]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,12]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8158, 6591, F81, 25) (dual of [6591, 6533, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8158, 6589, F81, 25) (dual of [6589, 6531, 26]-code), using
- net defined by OOA [i] based on linear OOA(8158, 549, F81, 25, 25) (dual of [(549, 25), 13667, 26]-NRT-code), using
(33, 57, 6593)-Net over F81 — Digital
Digital (33, 57, 6593)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8157, 6593, F81, 24) (dual of [6593, 6536, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(12) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(8110, 32, F81, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,81)), using
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- Reed–Solomon code RS(71,81) [i]
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- construction X applied to Ce(23) ⊂ Ce(12) [i] based on
(33, 57, large)-Net in Base 81 — Upper bound on s
There is no (33, 57, large)-net in base 81, because
- 22 times m-reduction [i] would yield (33, 35, large)-net in base 81, but