Best Known (187, 187+37, s)-Nets in Base 3
(187, 187+37, 1480)-Net over F3 — Constructive and digital
Digital (187, 224, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 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, 57, 370)-net over F81, using
(187, 187+37, 8793)-Net over F3 — Digital
Digital (187, 224, 8793)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3224, 8793, F3, 2, 37) (dual of [(8793, 2), 17362, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 9855, F3, 2, 37) (dual of [(9855, 2), 19486, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3224, 19710, F3, 37) (dual of [19710, 19486, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3224, 19710, F3, 37) (dual of [19710, 19486, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 9855, F3, 2, 37) (dual of [(9855, 2), 19486, 38]-NRT-code), using
(187, 187+37, 3076803)-Net in Base 3 — Upper bound on s
There is no (187, 224, 3076804)-net in base 3, because
- 1 times m-reduction [i] would yield (187, 223, 3076804)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25005 775688 335695 855024 009172 285607 991976 534048 368719 753223 514837 503791 822115 215697 835798 563900 348375 810745 > 3223 [i]