Best Known (34, 52, s)-Nets in Base 81
(34, 52, 59049)-Net over F81 — Constructive and digital
Digital (34, 52, 59049)-net over F81, using
- net defined by OOA [i] based on linear OOA(8152, 59049, F81, 18, 18) (dual of [(59049, 18), 1062830, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OA 9-folding and stacking [i] based on linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using
(34, 52, 184620)-Net over F81 — Digital
Digital (34, 52, 184620)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8152, 184620, F81, 2, 18) (dual of [(184620, 2), 369188, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 265722, F81, 2, 18) (dual of [(265722, 2), 531392, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8152, 531444, F81, 18) (dual of [531444, 531392, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(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(17) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(8152, 531444, F81, 18) (dual of [531444, 531392, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 265722, F81, 2, 18) (dual of [(265722, 2), 531392, 19]-NRT-code), using
(34, 52, large)-Net in Base 81 — Upper bound on s
There is no (34, 52, large)-net in base 81, because
- 16 times m-reduction [i] would yield (34, 36, large)-net in base 81, but