Best Known (107, 128, s)-Nets in Base 3
(107, 128, 1970)-Net over F3 — Constructive and digital
Digital (107, 128, 1970)-net over F3, using
- net defined by OOA [i] based on linear OOA(3128, 1970, F3, 21, 21) (dual of [(1970, 21), 41242, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3128, 19701, F3, 21) (dual of [19701, 19573, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3128, 19703, F3, 21) (dual of [19703, 19575, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3127, 19684, F3, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3128, 19703, F3, 21) (dual of [19703, 19575, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3128, 19701, F3, 21) (dual of [19701, 19573, 22]-code), using
(107, 128, 8242)-Net over F3 — Digital
Digital (107, 128, 8242)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 8242, F3, 2, 21) (dual of [(8242, 2), 16356, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3128, 9851, F3, 2, 21) (dual of [(9851, 2), 19574, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3128, 19702, F3, 21) (dual of [19702, 19574, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3128, 19703, F3, 21) (dual of [19703, 19575, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3127, 19684, F3, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3128, 19703, F3, 21) (dual of [19703, 19575, 22]-code), using
- OOA 2-folding [i] based on linear OA(3128, 19702, F3, 21) (dual of [19702, 19574, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(3128, 9851, F3, 2, 21) (dual of [(9851, 2), 19574, 22]-NRT-code), using
(107, 128, 2596478)-Net in Base 3 — Upper bound on s
There is no (107, 128, 2596479)-net in base 3, because
- 1 times m-reduction [i] would yield (107, 127, 2596479)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 930076 082077 885549 576621 954915 452100 504900 941390 669834 468589 > 3127 [i]