MinT - Best Known (230−41, 230, s)-Nets in Base 3

Best Known (230−41, 230, s)-Nets in Base 3

(230−41, 230, 1480)-Net over F₃ — Constructive and digital

Digital (189, 230, 1480)-net over F₃, using

3² times duplication [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]

(230−41, 230, 4838)-Net over F₃ — Digital

Digital (189, 230, 4838)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁰, 4838, F₃, 41) (dual of [4838, 4608, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁰, 6606, F₃, 41) (dual of [6606, 6376, 42]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²²⁸, 6604, F₃, 41) (dual of [6604, 6376, 42]-code), using
    - constructionÂ X applied to C_e(40) âŠ‚ C_e(34) [i] based on
      1. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      2. linear OA(3¹⁸⁵, 6561, F₃, 35) (dual of [6561, 6376, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      3. linear OA(3¹¹, 43, F₃, 5) (dual of [43, 32, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(230−41, 230, 1205889)-Net in Base 3 — Upper bound on s

There is no (189, 230, 1205890)-net in base 3, because

1 times m-reduction [i] would yield (189, 229, 1205890)-net in base 3, but
- 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]