Best Known (181−31, 181, s)-Nets in Base 3
(181−31, 181, 1312)-Net over F3 — Constructive and digital
Digital (150, 181, 1312)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 1312, F3, 31, 31) (dual of [(1312, 31), 40491, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3181, 19681, F3, 31) (dual of [19681, 19500, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3181, 19681, F3, 31) (dual of [19681, 19500, 32]-code), using
(181−31, 181, 6561)-Net over F3 — Digital
Digital (150, 181, 6561)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 6561, F3, 3, 31) (dual of [(6561, 3), 19502, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 3-folding [i] based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
(181−31, 181, 1706821)-Net in Base 3 — Upper bound on s
There is no (150, 181, 1706822)-net in base 3, because
- 1 times m-reduction [i] would yield (150, 180, 1706822)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 76 177926 277976 006557 807160 974387 816933 251335 918347 158269 628880 528979 093817 900321 305801 > 3180 [i]