Best Known (87, 87+18, s)-Nets in Base 3
(87, 87+18, 733)-Net over F3 — Constructive and digital
Digital (87, 105, 733)-net over F3, using
- net defined by OOA [i] based on linear OOA(3105, 733, F3, 18, 18) (dual of [(733, 18), 13089, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3105, 6597, F3, 18) (dual of [6597, 6492, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(365, 6562, F3, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3105, 6597, F3, 18) (dual of [6597, 6492, 19]-code), using
(87, 87+18, 4278)-Net over F3 — Digital
Digital (87, 105, 4278)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3105, 4278, F3, 18) (dual of [4278, 4173, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(365, 6562, F3, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
(87, 87+18, 764066)-Net in Base 3 — Upper bound on s
There is no (87, 105, 764067)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 125 237213 643704 896681 991064 597388 376621 242116 220663 > 3105 [i]