Best Known (165, 165+34, s)-Nets in Base 3
(165, 165+34, 1157)-Net over F3 — Constructive and digital
Digital (165, 199, 1157)-net over F3, using
- net defined by OOA [i] based on linear OOA(3199, 1157, F3, 34, 34) (dual of [(1157, 34), 39139, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(3199, 19669, F3, 34) (dual of [19669, 19470, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(3199, 19669, F3, 34) (dual of [19669, 19470, 35]-code), using
(165, 165+34, 6654)-Net over F3 — Digital
Digital (165, 199, 6654)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3199, 6654, F3, 2, 34) (dual of [(6654, 2), 13109, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3199, 9841, F3, 2, 34) (dual of [(9841, 2), 19483, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3199, 19682, F3, 34) (dual of [19682, 19483, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- OOA 2-folding [i] based on linear OA(3199, 19682, F3, 34) (dual of [19682, 19483, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3199, 9841, F3, 2, 34) (dual of [(9841, 2), 19483, 35]-NRT-code), using
(165, 165+34, 1380496)-Net in Base 3 — Upper bound on s
There is no (165, 199, 1380497)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 88538 588852 450885 256575 924513 422715 038850 268409 283633 031670 573559 050009 976729 072966 876943 729891 > 3199 [i]