Best Known (48, 72, s)-Nets in Base 81
(48, 72, 44287)-Net over F81 — Constructive and digital
Digital (48, 72, 44287)-net over F81, using
- 812 times duplication [i] based on digital (46, 70, 44287)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 44287, F81, 24, 24) (dual of [(44287, 24), 1062818, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [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(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- net defined by OOA [i] based on linear OOA(8170, 44287, F81, 24, 24) (dual of [(44287, 24), 1062818, 25]-NRT-code), using
(48, 72, 249461)-Net over F81 — Digital
Digital (48, 72, 249461)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8172, 249461, F81, 2, 24) (dual of [(249461, 2), 498850, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 265726, F81, 2, 24) (dual of [(265726, 2), 531380, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8172, 531452, F81, 24) (dual of [531452, 531380, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [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(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(8172, 531452, F81, 24) (dual of [531452, 531380, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 265726, F81, 2, 24) (dual of [(265726, 2), 531380, 25]-NRT-code), using
(48, 72, large)-Net in Base 81 — Upper bound on s
There is no (48, 72, large)-net in base 81, because
- 22 times m-reduction [i] would yield (48, 50, large)-net in base 81, but