Best Known (3, 3+4, s)-Nets in Base 81
(3, 3+4, 3281)-Net over F81 — Constructive and digital
Digital (3, 7, 3281)-net over F81, using
- net defined by OOA [i] based on linear OOA(817, 3281, F81, 4, 4) (dual of [(3281, 4), 13117, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(817, 6562, F81, 4) (dual of [6562, 6555, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(817, 6562, F81, 4) (dual of [6562, 6555, 5]-code), using
(3, 3+4, 6563)-Net over F81 — Digital
Digital (3, 7, 6563)-net over F81, using
- net defined by OOA [i] based on linear OOA(817, 6563, F81, 4, 4) (dual of [(6563, 4), 26245, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(817, 6563, F81, 3, 4) (dual of [(6563, 3), 19682, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- appending kth column [i] based on linear OOA(817, 6563, F81, 3, 4) (dual of [(6563, 3), 19682, 5]-NRT-code), using
(3, 3+4, 84551)-Net in Base 81 — Upper bound on s
There is no (3, 7, 84552)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 22 877214 347521 > 817 [i]