MinT - Best Known (177, 218, s)-Nets in Base 3

Best Known (177, 218, s)-Nets in Base 3

(177, 218, 896)-Net over F₃ — Constructive and digital

Digital (177, 218, 896)-net over F₃, using

3² times duplication [i] based on digital (175, 216, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 54, 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]

(177, 218, 3440)-Net over F₃ — Digital

Digital (177, 218, 3440)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁸, 3440, F₃, 41) (dual of [3440, 3222, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁸, 6570, F₃, 41) (dual of [6570, 6352, 42]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3²¹⁷, 6569, F₃, 41) (dual of [6569, 6352, 42]-code), using
    - constructionÂ X applied to C_e(40) âŠ‚ C_e(39) [i] based on
      1. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      2. linear OA(3²⁰⁹, 6561, F₃, 40) (dual of [6561, 6352, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
      3. linear OA(3⁰, 8, F₃, 0) (dual of [8, 8, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(177, 218, 623775)-Net in Base 3 — Upper bound on s

There is no (177, 218, 623776)-net in base 3, because

1 times m-reduction [i] would yield (177, 217, 623776)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 302047 369340 383904 636796 288694 131812 965661 797937 198831 792737 009252 721599 719731 707441 262020 013390 529281 > 3²¹⁷ [i]