Best Known (164, 164+24, s)-Nets in Base 3
(164, 164+24, 14766)-Net over F3 — Constructive and digital
Digital (164, 188, 14766)-net over F3, using
- t-expansion [i] based on digital (163, 188, 14766)-net over F3, using
- net defined by OOA [i] based on linear OOA(3188, 14766, F3, 25, 25) (dual of [(14766, 25), 368962, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3188, 177193, F3, 25) (dual of [177193, 177005, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 177203, F3, 25) (dual of [177203, 177015, 26]-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(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-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(3188, 177203, F3, 25) (dual of [177203, 177015, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3188, 177193, F3, 25) (dual of [177193, 177005, 26]-code), using
- net defined by OOA [i] based on linear OOA(3188, 14766, F3, 25, 25) (dual of [(14766, 25), 368962, 26]-NRT-code), using
(164, 164+24, 72992)-Net over F3 — Digital
Digital (164, 188, 72992)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3188, 72992, F3, 2, 24) (dual of [(72992, 2), 145796, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 88601, F3, 2, 24) (dual of [(88601, 2), 177014, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(3188, 88601, F3, 2, 25) (dual of [(88601, 2), 177014, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3188, 177202, F3, 25) (dual of [177202, 177014, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 177203, F3, 25) (dual of [177203, 177015, 26]-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(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-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(3188, 177203, F3, 25) (dual of [177203, 177015, 26]-code), using
- OOA 2-folding [i] based on linear OA(3188, 177202, F3, 25) (dual of [177202, 177014, 26]-code), using
- strength reduction [i] based on linear OOA(3188, 88601, F3, 2, 25) (dual of [(88601, 2), 177014, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 88601, F3, 2, 24) (dual of [(88601, 2), 177014, 25]-NRT-code), using
(164, 164+24, large)-Net in Base 3 — Upper bound on s
There is no (164, 188, large)-net in base 3, because
- 22 times m-reduction [i] would yield (164, 166, large)-net in base 3, but