MinT - Best Known (229−40, 229, s)-Nets in Base 3

Best Known (229−40, 229, s)-Nets in Base 3

(229−40, 229, 1480)-Net over F₃ — Constructive and digital

Digital (189, 229, 1480)-net over F₃, using

3¹ times duplication [i] based on digital (188, 228, 1480)-net over F₃, using
- t-expansion [i] based on digital (187, 228, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 57, 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]

(229−40, 229, 5440)-Net over F₃ — Digital

Digital (189, 229, 5440)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁹, 5440, F₃, 40) (dual of [5440, 5211, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁹, 6582, F₃, 40) (dual of [6582, 6353, 41]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3²⁰, 21, F₃, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,3)), using
      - dual of repetition code with length 21 [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]

(229−40, 229, 1205889)-Net in Base 3 — Upper bound on s

There is no (189, 229, 1205890)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 229284 597602 308501 564778 192863 818042 082691 042923 850182 756542 526418 637466 391751 574618 186142 753374 177114 084809 > 3²²⁹ [i]