MinT - Best Known (190, 190+35, s)-Nets in Base 3

Best Known (190, 190+35, s)-Nets in Base 3

(190, 190+35, 1492)-Net over F₃ — Constructive and digital

Digital (190, 225, 1492)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (169, 204, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 51, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(190, 190+35, 11370)-Net over F₃ — Digital

Digital (190, 225, 11370)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁵, 11370, F₃, 35) (dual of [11370, 11145, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁵, 19701, F₃, 35) (dual of [19701, 19476, 36]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁷, 18, F₃, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,3)), using
      - dual of repetition code with length 18 [i]
    2. linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using
      - an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(190, 190+35, 6945257)-Net in Base 3 — Upper bound on s

There is no (190, 225, 6945258)-net in base 3, because

1 times m-reduction [i] would yield (190, 224, 6945258)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75017 306843 928922 860244 723833 620770 823082 324608 581566 837378 742006 986109 064347 569097 152034 348906 392096 024117 > 3²²⁴ [i]