Best Known (223−36, 223, s)-Nets in Base 3
(223−36, 223, 1480)-Net over F3 — Constructive and digital
Digital (187, 223, 1480)-net over F3, using
- 5 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
(223−36, 223, 9858)-Net over F3 — Digital
Digital (187, 223, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3223, 9858, F3, 2, 36) (dual of [(9858, 2), 19493, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3223, 19716, F3, 36) (dual of [19716, 19493, 37]-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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3223, 19716, F3, 36) (dual of [19716, 19493, 37]-code), using
(223−36, 223, 3076803)-Net in Base 3 — Upper bound on s
There is no (187, 223, 3076804)-net in base 3, because
- 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]