Best Known (47, 69, s)-Nets in Base 81
(47, 69, 48314)-Net over F81 — Constructive and digital
Digital (47, 69, 48314)-net over F81, using
- 1 times m-reduction [i] based on digital (47, 70, 48314)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 48314, F81, 23, 23) (dual of [(48314, 23), 1111152, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8170, 531455, F81, 23) (dual of [531455, 531385, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 531457, F81, 23) (dual of [531457, 531387, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 531457, F81, 23) (dual of [531457, 531387, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8170, 531455, F81, 23) (dual of [531455, 531385, 24]-code), using
- net defined by OOA [i] based on linear OOA(8170, 48314, F81, 23, 23) (dual of [(48314, 23), 1111152, 24]-NRT-code), using
(47, 69, 319928)-Net over F81 — Digital
Digital (47, 69, 319928)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8169, 319928, F81, 22) (dual of [319928, 319859, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 531464, F81, 22) (dual of [531464, 531395, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(8169, 531464, F81, 22) (dual of [531464, 531395, 23]-code), using
(47, 69, large)-Net in Base 81 — Upper bound on s
There is no (47, 69, large)-net in base 81, because
- 20 times m-reduction [i] would yield (47, 49, large)-net in base 81, but