MinT - Best Known (189, 189+35, s)-Nets in Base 3

Best Known (189, 189+35, s)-Nets in Base 3

(189, 189+35, 1490)-Net over F₃ — Constructive and digital

Digital (189, 224, 1490)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, 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]

(189, 189+35, 10997)-Net over F₃ — Digital

Digital (189, 224, 10997)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁴, 10997, F₃, 35) (dual of [10997, 10773, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁴, 19744, F₃, 35) (dual of [19744, 19520, 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(3¹⁶, 61, F₃, 6) (dual of [61, 45, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 80, F₃, 6) (dual of [80, 64, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(189, 189+35, 6510619)-Net in Base 3 — Upper bound on s

There is no (189, 224, 6510620)-net in base 3, because

1 times m-reduction [i] would yield (189, 223, 6510620)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25005 758792 184611 351286 133697 209229 201443 776892 666646 758200 541090 850003 152351 084635 686882 991472 594760 552505 > 3²²³ [i]