Best Known (81−27, 81, s)-Nets in Base 81
(81−27, 81, 40880)-Net over F81 — Constructive and digital
Digital (54, 81, 40880)-net over F81, using
- 812 times duplication [i] based on digital (52, 79, 40880)-net over F81, using
- net defined by OOA [i] based on linear OOA(8179, 40880, F81, 27, 27) (dual of [(40880, 27), 1103681, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8179, 531441, F81, 27) (dual of [531441, 531362, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- OOA 13-folding and stacking with additional row [i] based on linear OA(8179, 531441, F81, 27) (dual of [531441, 531362, 28]-code), using
- net defined by OOA [i] based on linear OOA(8179, 40880, F81, 27, 27) (dual of [(40880, 27), 1103681, 28]-NRT-code), using
(81−27, 81, 234611)-Net over F81 — Digital
Digital (54, 81, 234611)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8181, 234611, F81, 2, 27) (dual of [(234611, 2), 469141, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8181, 265726, F81, 2, 27) (dual of [(265726, 2), 531371, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8181, 531452, F81, 27) (dual of [531452, 531371, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(8179, 531441, F81, 27) (dual of [531441, 531362, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(812, 11, F81, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(8181, 531452, F81, 27) (dual of [531452, 531371, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(8181, 265726, F81, 2, 27) (dual of [(265726, 2), 531371, 28]-NRT-code), using
(81−27, 81, large)-Net in Base 81 — Upper bound on s
There is no (54, 81, large)-net in base 81, because
- 25 times m-reduction [i] would yield (54, 56, large)-net in base 81, but