MinT - Best Known (174−31, 174, s)-Nets in Base 3

Best Known (174−31, 174, s)-Nets in Base 3

(174−31, 174, 704)-Net over F₃ — Constructive and digital

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]

(174−31, 174, 4069)-Net over F₃ — Digital

Digital (143, 174, 4069)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁴, 4069, F₃, 31) (dual of [4069, 3895, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁷⁴, 6607, F₃, 31) (dual of [6607, 6433, 32]-code), using
  - 2 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]

(174−31, 174, 1022183)-Net in Base 3 — Upper bound on s

There is no (143, 174, 1022184)-net in base 3, because

1 times m-reduction [i] would yield (143, 173, 1022184)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34832 040219 294961 161547 678549 944448 707966 292036 092684 453245 738487 062334 791715 823841 > 3¹⁷³ [i]