Best Known (37, 53, s)-Nets in Base 81
(37, 53, 66434)-Net over F81 — Constructive and digital
Digital (37, 53, 66434)-net over F81, using
- net defined by OOA [i] based on linear OOA(8153, 66434, F81, 16, 16) (dual of [(66434, 16), 1062891, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8153, 531472, F81, 16) (dual of [531472, 531419, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(7) [i] based on
- 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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(817, 31, F81, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to Ce(15) ⊂ Ce(7) [i] based on
- OA 8-folding and stacking [i] based on linear OA(8153, 531472, F81, 16) (dual of [531472, 531419, 17]-code), using
(37, 53, 531472)-Net over F81 — Digital
Digital (37, 53, 531472)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8153, 531472, F81, 16) (dual of [531472, 531419, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(7) [i] based on
- 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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(817, 31, F81, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to Ce(15) ⊂ Ce(7) [i] based on
(37, 53, large)-Net in Base 81 — Upper bound on s
There is no (37, 53, large)-net in base 81, because
- 14 times m-reduction [i] would yield (37, 39, large)-net in base 81, but