MinT - Best Known (175, 175+37, s)-Nets in Base 3

Best Known (175, 175+37, s)-Nets in Base 3

(175, 175+37, 1480)-Net over F₃ — Constructive and digital

Digital (175, 212, 1480)-net over F₃, using

trace code for nets [i] based on digital (16, 53, 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]

(175, 175+37, 5198)-Net over F₃ — Digital

Digital (175, 212, 5198)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹², 5198, F₃, 37) (dual of [5198, 4986, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹², 6620, F₃, 37) (dual of [6620, 6408, 38]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3²⁰⁹, 6617, F₃, 37) (dual of [6617, 6408, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(28) [i] based on
      1. linear OA(3¹⁹³, 6561, F₃, 37) (dual of [6561, 6368, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(175, 175+37, 1479163)-Net in Base 3 — Upper bound on s

There is no (175, 212, 1479164)-net in base 3, because

1 times m-reduction [i] would yield (175, 211, 1479164)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47052 781134 534152 699837 131693 728196 235828 645386 098159 001887 075690 233584 125663 160140 343443 798961 943305 > 3²¹¹ [i]