Best Known (186−24, 186, s)-Nets in Base 3
(186−24, 186, 14766)-Net over F3 — Constructive and digital
Digital (162, 186, 14766)-net over F3, using
- net defined by OOA [i] based on linear OOA(3186, 14766, F3, 24, 24) (dual of [(14766, 24), 354198, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3186, 177192, F3, 24) (dual of [177192, 177006, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 177201, F3, 24) (dual of [177201, 177015, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 53, F3, 4) (dual of [53, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3186, 177201, F3, 24) (dual of [177201, 177015, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3186, 177192, F3, 24) (dual of [177192, 177006, 25]-code), using
(186−24, 186, 65739)-Net over F3 — Digital
Digital (162, 186, 65739)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 65739, F3, 2, 24) (dual of [(65739, 2), 131292, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 88600, F3, 2, 24) (dual of [(88600, 2), 177014, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3186, 177200, F3, 24) (dual of [177200, 177014, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 177201, F3, 24) (dual of [177201, 177015, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 53, F3, 4) (dual of [53, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3186, 177201, F3, 24) (dual of [177201, 177015, 25]-code), using
- OOA 2-folding [i] based on linear OA(3186, 177200, F3, 24) (dual of [177200, 177014, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 88600, F3, 2, 24) (dual of [(88600, 2), 177014, 25]-NRT-code), using
(186−24, 186, large)-Net in Base 3 — Upper bound on s
There is no (162, 186, large)-net in base 3, because
- 22 times m-reduction [i] would yield (162, 164, large)-net in base 3, but