MinT - Best Known (172, 172+35, s)-Nets in Base 3

Best Known (172, 172+35, s)-Nets in Base 3

(172, 172+35, 1480)-Net over F₃ — Constructive and digital

Digital (172, 207, 1480)-net over F₃, using

1 times m-reduction [i] based on digital (172, 208, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 52, 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]

(172, 172+35, 6230)-Net over F₃ — Digital

Digital (172, 207, 6230)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁷, 6230, F₃, 35) (dual of [6230, 6023, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁷, 9841, F₃, 35) (dual of [9841, 9634, 36]-code), using
  - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,3,5,…,69}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(172, 172+35, 2170195)-Net in Base 3 — Upper bound on s

There is no (172, 207, 2170196)-net in base 3, because

1 times m-reduction [i] would yield (172, 206, 2170196)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 193 633208 252582 078676 396497 024517 735803 982673 972827 208006 682748 634785 026168 217767 055884 703796 693929 > 3²⁰⁶ [i]