MinT - Best Known (94, 117, s)-Nets in Base 3

Best Known (94, 117, s)-Nets in Base 3

(94, 117, 640)-Net over F₃ — Constructive and digital

Digital (94, 117, 640)-net over F₃, using

3¹ times duplication [i] based on digital (93, 116, 640)-net over F₃, using
- t-expansion [i] based on digital (92, 116, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 29, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(94, 117, 1855)-Net over F₃ — Digital

Digital (94, 117, 1855)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁷, 1855, F₃, 23) (dual of [1855, 1738, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁷, 2199, F₃, 23) (dual of [2199, 2082, 24]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹¹, 12, F₃, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
      - dual of repetition code with length 12 [i]
    2. linear OA(3¹⁰⁶, 2187, F₃, 23) (dual of [2187, 2081, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(94, 117, 263892)-Net in Base 3 — Upper bound on s

There is no (94, 117, 263893)-net in base 3, because

1 times m-reduction [i] would yield (94, 116, 263893)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 185584 837668 934301 250110 510350 098361 906932 475077 786059 > 3¹¹⁶ [i]