Best Known (178, 216, s)-Nets in Base 3
(178, 216, 1480)-Net over F3 — Constructive and digital
Digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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
(178, 216, 5014)-Net over F3 — Digital
Digital (178, 216, 5014)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 5014, F3, 38) (dual of [5014, 4798, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 6616, F3, 38) (dual of [6616, 6400, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(315, 55, F3, 6) (dual of [55, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 6616, F3, 38) (dual of [6616, 6400, 39]-code), using
(178, 216, 1052672)-Net in Base 3 — Upper bound on s
There is no (178, 216, 1052673)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 433955 655037 344929 840498 129303 115155 026754 057548 589076 540710 691186 313409 716205 884637 882645 470489 506779 > 3216 [i]