MinT - Best Known (225−38, 225, s)-Nets in Base 3

Best Known (225−38, 225, s)-Nets in Base 3

(225−38, 225, 1480)-Net over F₃ — Constructive and digital

Digital (187, 225, 1480)-net over F₃, using

3 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
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(225−38, 225, 6610)-Net over F₃ — Digital

Digital (187, 225, 6610)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁵, 6610, F₃, 38) (dual of [6610, 6385, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁵, 9841, F₃, 38) (dual of [9841, 9616, 39]-code), using
  - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,3,5,…,75}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]

(225−38, 225, 1771336)-Net in Base 3 — Upper bound on s

There is no (187, 225, 1771337)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 225052 669372 611556 009131 423718 172163 816362 928565 546962 263263 184750 097403 067647 077537 341558 900700 134144 155707 > 3²²⁵ [i]