MinT - Best Known (175−31, 175, s)-Nets in Base 3

Best Known (175−31, 175, s)-Nets in Base 3

(175−31, 175, 704)-Net over F₃ — Constructive and digital

Digital (144, 175, 704)-net over F₃, using

3¹ times duplication [i] based on digital (143, 174, 704)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (121, 152, 688)-net over F₃, using
    - trace code for nets [i] based on digital (7, 38, 172)-net over F₈₁, using
      - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(175−31, 175, 4227)-Net over F₃ — Digital

Digital (144, 175, 4227)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁵, 4227, F₃, 31) (dual of [4227, 4052, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁷⁵, 6608, F₃, 31) (dual of [6608, 6433, 32]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3¹⁷², 6605, F₃, 31) (dual of [6605, 6433, 32]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      2. linear OA(3¹²⁹, 6562, F₃, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [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]

(175−31, 175, 1099860)-Net in Base 3 — Upper bound on s

There is no (144, 175, 1099861)-net in base 3, because

1 times m-reduction [i] would yield (144, 174, 1099861)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104496 861151 028039 822755 591204 683904 989150 876106 532770 826186 455385 509896 686295 426491 > 3¹⁷⁴ [i]