Best Known (156−25, 156, s)-Nets in Base 3
(156−25, 156, 1644)-Net over F3 — Constructive and digital
Digital (131, 156, 1644)-net over F3, using
- net defined by OOA [i] based on linear OOA(3156, 1644, F3, 25, 25) (dual of [(1644, 25), 40944, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3156, 19729, F3, 25) (dual of [19729, 19573, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3156, 19731, F3, 25) (dual of [19731, 19575, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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(311, 47, F3, 5) (dual of [47, 36, 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(3156, 19731, F3, 25) (dual of [19731, 19575, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3156, 19729, F3, 25) (dual of [19729, 19573, 26]-code), using
(156−25, 156, 9865)-Net over F3 — Digital
Digital (131, 156, 9865)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3156, 9865, F3, 2, 25) (dual of [(9865, 2), 19574, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3156, 19730, F3, 25) (dual of [19730, 19574, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3156, 19731, F3, 25) (dual of [19731, 19575, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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(311, 47, F3, 5) (dual of [47, 36, 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(3156, 19731, F3, 25) (dual of [19731, 19575, 26]-code), using
- OOA 2-folding [i] based on linear OA(3156, 19730, F3, 25) (dual of [19730, 19574, 26]-code), using
(156−25, 156, 3847213)-Net in Base 3 — Upper bound on s
There is no (131, 156, 3847214)-net in base 3, because
- 1 times m-reduction [i] would yield (131, 155, 3847214)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 89 907246 179214 732581 812686 147696 903019 124435 572532 736327 565431 572753 349849 > 3155 [i]