MinT - Best Known (50−15, 50, s)-Nets in Base 3

Best Known (50−15, 50, s)-Nets in Base 3

(50−15, 50, 114)-Net over F₃ — Constructive and digital

Digital (35, 50, 114)-net over F₃, using

1 times m-reduction [i] based on digital (35, 51, 114)-net over F₃, using
- trace code for nets [i] based on digital (1, 17, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(50−15, 50, 168)-Net over F₃ — Digital

Digital (35, 50, 168)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁰, 168, F₃, 15) (dual of [168, 118, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁰, 242, F₃, 15) (dual of [242, 192, 16]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(50−15, 50, 3689)-Net in Base 3 — Upper bound on s

There is no (35, 50, 3690)-net in base 3, because

1 times m-reduction [i] would yield (35, 49, 3690)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 239502 726051 030177 508297 > 3⁴⁹ [i]