MinT - Best Known (140−31, 140, s)-Nets in Base 3

Best Known (140−31, 140, s)-Nets in Base 3

(140−31, 140, 600)-Net over F₃ — Constructive and digital

Digital (109, 140, 600)-net over F₃, using

trace code for nets [i] based on digital (4, 35, 150)-net over F₈₁, using
- net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(140−31, 140, 1093)-Net over F₃ — Digital

Digital (109, 140, 1093)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁰, 1093, F₃, 31) (dual of [1093, 953, 32]-code), using
- the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,57,113,…,−505}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(140−31, 140, 84720)-Net in Base 3 — Upper bound on s

There is no (109, 140, 84721)-net in base 3, because

1 times m-reduction [i] would yield (109, 139, 84721)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 088965 130222 806268 704257 240262 240830 632483 332572 148495 340896 987691 > 3¹³⁹ [i]