Best Known (52, 52+24, s)-Nets in Base 81
(52, 52+24, 44289)-Net over F81 — Constructive and digital
Digital (52, 76, 44289)-net over F81, using
- net defined by OOA [i] based on linear OOA(8176, 44289, F81, 24, 24) (dual of [(44289, 24), 1062860, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8176, 531468, F81, 24) (dual of [531468, 531392, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(16) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-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(23) ⊂ Ce(16) [i] based on
- OA 12-folding and stacking [i] based on linear OA(8176, 531468, F81, 24) (dual of [531468, 531392, 25]-code), using
(52, 52+24, 363036)-Net over F81 — Digital
Digital (52, 76, 363036)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8176, 363036, F81, 24) (dual of [363036, 362960, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8176, 531468, F81, 24) (dual of [531468, 531392, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(16) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-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(23) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8176, 531468, F81, 24) (dual of [531468, 531392, 25]-code), using
(52, 52+24, large)-Net in Base 81 — Upper bound on s
There is no (52, 76, large)-net in base 81, because
- 22 times m-reduction [i] would yield (52, 54, large)-net in base 81, but