MinT - Best Known (222−35, 222, s)-Nets in Base 3

Best Known (222−35, 222, s)-Nets in Base 3

(222−35, 222, 1487)-Net over F₃ — Constructive and digital

Digital (187, 222, 1487)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, 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]

(222−35, 222, 10286)-Net over F₃ — Digital

Digital (187, 222, 10286)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²², 10286, F₃, 35) (dual of [10286, 10064, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²², 19736, F₃, 35) (dual of [19736, 19514, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(27) [i] based on
    1. 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]
    2. linear OA(3¹⁶³, 19683, F₃, 28) (dual of [19683, 19520, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
      - codes constructed by inverting constructionÂ Y₁ [i]

(222−35, 222, 5721241)-Net in Base 3 — Upper bound on s

There is no (187, 222, 5721242)-net in base 3, because

1 times m-reduction [i] would yield (187, 221, 5721242)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2778 421380 842358 140963 267787 872744 285250 677786 723740 438841 392052 496674 478913 463509 986484 573737 523529 771925 > 3²²¹ [i]