MinT - Best Known (112−25, 112, s)-Nets in Base 3

Best Known (112−25, 112, s)-Nets in Base 3

(112−25, 112, 464)-Net over F₃ — Constructive and digital

Digital (87, 112, 464)-net over F₃, using

t-expansion [i] based on digital (86, 112, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 28, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(112−25, 112, 926)-Net over F₃ — Digital

Digital (87, 112, 926)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹², 926, F₃, 25) (dual of [926, 814, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹², 1093, F₃, 25) (dual of [1093, 981, 26]-code), using
  - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,57,113,…,252}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(112−25, 112, 68490)-Net in Base 3 — Upper bound on s

There is no (87, 112, 68491)-net in base 3, because

1 times m-reduction [i] would yield (87, 111, 68491)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 91306 139051 283437 463701 728599 767984 394020 713564 657353 > 3¹¹¹ [i]