Best Known (59, 59+14, s)-Nets in Base 3
(59, 59+14, 938)-Net over F3 — Constructive and digital
Digital (59, 73, 938)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 938, F3, 14, 14) (dual of [(938, 14), 13059, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
(59, 59+14, 2938)-Net over F3 — Digital
Digital (59, 73, 2938)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(373, 2938, F3, 2, 14) (dual of [(2938, 2), 5803, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(373, 3284, F3, 2, 14) (dual of [(3284, 2), 6495, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(373, 6568, F3, 14) (dual of [6568, 6495, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- OOA 2-folding [i] based on linear OA(373, 6568, F3, 14) (dual of [6568, 6495, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(373, 3284, F3, 2, 14) (dual of [(3284, 2), 6495, 15]-NRT-code), using
(59, 59+14, 159795)-Net in Base 3 — Upper bound on s
There is no (59, 73, 159796)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 67587 193944 930453 294519 181655 121969 > 373 [i]