MinT - Best Known (180−31, 180, s)-Nets in Base 3

Best Known (180−31, 180, s)-Nets in Base 3

(180−31, 180, 896)-Net over F₃ — Constructive and digital

Digital (149, 180, 896)-net over F₃, using

t-expansion [i] based on digital (148, 180, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 45, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(180−31, 180, 5114)-Net over F₃ — Digital

Digital (149, 180, 5114)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁰, 5114, F₃, 31) (dual of [5114, 4934, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, 9841, F₃, 31) (dual of [9841, 9661, 32]-code), using
  - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,1313,2625,…,−3}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(180−31, 180, 1586279)-Net in Base 3 — Upper bound on s

There is no (149, 180, 1586280)-net in base 3, because

1 times m-reduction [i] would yield (149, 179, 1586280)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 392625 077236 407331 127728 157726 725341 909979 551520 208136 824010 774520 306210 901255 670497 > 3¹⁷⁹ [i]