MinT - Best Known (183, 218, s)-Nets in Base 3

Best Known (183, 218, s)-Nets in Base 3

(183, 218, 1480)-Net over F₃ — Constructive and digital

Digital (183, 218, 1480)-net over F₃, using

t-expansion [i] based on digital (181, 218, 1480)-net over F₃, using
- 2 times m-reduction [i] based on digital (181, 220, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 55, 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]

(183, 218, 9861)-Net over F₃ — Digital

Digital (183, 218, 9861)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁸, 9861, F₃, 2, 35) (dual of [(9861, 2), 19504, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²¹⁸, 19722, F₃, 35) (dual of [19722, 19504, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(28) [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(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
      - generator matrices provided on Edelâ€™s homepage [i]

(183, 218, 4418003)-Net in Base 3 — Upper bound on s

There is no (183, 218, 4418004)-net in base 3, because

1 times m-reduction [i] would yield (183, 217, 4418004)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 301472 590282 302796 523772 651358 885898 461020 308901 860114 848433 897552 209952 832178 700720 794125 691763 307689 > 3²¹⁷ [i]