Best Known (41, 41+20, s)-Nets in Base 81
(41, 41+20, 53145)-Net over F81 — Constructive and digital
Digital (41, 61, 53145)-net over F81, using
- 811 times duplication [i] based on digital (40, 60, 53145)-net over F81, using
- net defined by OOA [i] based on linear OOA(8160, 53145, F81, 20, 20) (dual of [(53145, 20), 1062840, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8160, 531450, F81, 20) (dual of [531450, 531390, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8160, 531452, F81, 20) (dual of [531452, 531392, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8160, 531452, F81, 20) (dual of [531452, 531392, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8160, 531450, F81, 20) (dual of [531450, 531390, 21]-code), using
- net defined by OOA [i] based on linear OOA(8160, 53145, F81, 20, 20) (dual of [(53145, 20), 1062840, 21]-NRT-code), using
(41, 41+20, 265728)-Net over F81 — Digital
Digital (41, 61, 265728)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8161, 265728, F81, 2, 20) (dual of [(265728, 2), 531395, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8161, 531456, F81, 20) (dual of [531456, 531395, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(8161, 531456, F81, 20) (dual of [531456, 531395, 21]-code), using
(41, 41+20, large)-Net in Base 81 — Upper bound on s
There is no (41, 61, large)-net in base 81, because
- 18 times m-reduction [i] would yield (41, 43, large)-net in base 81, but