Best Known (169, 169+34, s)-Nets in Base 3
(169, 169+34, 1480)-Net over F3 — Constructive and digital
Digital (169, 203, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
(169, 169+34, 7672)-Net over F3 — Digital
Digital (169, 203, 7672)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3203, 7672, F3, 2, 34) (dual of [(7672, 2), 15141, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3203, 9852, F3, 2, 34) (dual of [(9852, 2), 19501, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3203, 19704, F3, 34) (dual of [19704, 19501, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3203, 19705, F3, 34) (dual of [19705, 19502, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] 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]
- 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]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3203, 19705, F3, 34) (dual of [19705, 19502, 35]-code), using
- OOA 2-folding [i] based on linear OA(3203, 19704, F3, 34) (dual of [19704, 19501, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3203, 9852, F3, 2, 34) (dual of [(9852, 2), 19501, 35]-NRT-code), using
(169, 169+34, 1787723)-Net in Base 3 — Upper bound on s
There is no (169, 203, 1787724)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7 171624 661979 861722 764476 081930 899853 788435 407370 308813 343658 235805 079736 409342 818175 387942 646937 > 3203 [i]