Best Known (49−17, 49, s)-Nets in Base 81
(49−17, 49, 66430)-Net over F81 — Constructive and digital
Digital (32, 49, 66430)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 66430, F81, 17, 17) (dual of [(66430, 17), 1129261, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on 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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using
(49−17, 49, 192935)-Net over F81 — Digital
Digital (32, 49, 192935)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8149, 192935, F81, 2, 17) (dual of [(192935, 2), 385821, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8149, 265722, F81, 2, 17) (dual of [(265722, 2), 531395, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(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(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(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(8149, 265722, F81, 2, 17) (dual of [(265722, 2), 531395, 18]-NRT-code), using
(49−17, 49, large)-Net in Base 81 — Upper bound on s
There is no (32, 49, large)-net in base 81, because
- 15 times m-reduction [i] would yield (32, 34, large)-net in base 81, but