Best Known (33, 33+8, s)-Nets in Base 3
(33, 33+8, 1642)-Net over F3 — Constructive and digital
Digital (33, 41, 1642)-net over F3, using
- net defined by OOA [i] based on linear OOA(341, 1642, F3, 8, 8) (dual of [(1642, 8), 13095, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
(33, 33+8, 3284)-Net over F3 — Digital
Digital (33, 41, 3284)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(341, 3284, F3, 2, 8) (dual of [(3284, 2), 6527, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- OOA 2-folding [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
(33, 33+8, 85999)-Net in Base 3 — Upper bound on s
There is no (33, 41, 86000)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 36 473147 450661 648001 > 341 [i]