MinT - Best Known (174, 174+32, s)-Nets in Base 3

Best Known (174, 174+32, s)-Nets in Base 3

(174, 174+32, 1480)-Net over F₃ — Constructive and digital

Digital (174, 206, 1480)-net over F₃, using

t-expansion [i] based on digital (172, 206, 1480)-net over F₃, using
- 2 times m-reduction [i] based on digital (172, 208, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 52, 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]

(174, 174+32, 10938)-Net over F₃ — Digital

Digital (174, 206, 10938)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁶, 10938, F₃, 32) (dual of [10938, 10732, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁶, 19700, F₃, 32) (dual of [19700, 19494, 33]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁶, 17, F₃, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,3)), using
      - dual of repetition code with length 17 [i]
    2. linear OA(3¹⁹⁰, 19683, F₃, 32) (dual of [19683, 19493, 33]-code), using
      - an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(174, 174+32, 4725453)-Net in Base 3 — Upper bound on s

There is no (174, 206, 4725454)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 193 633175 308265 066813 744181 627537 704061 503664 368065 397468 530864 640308 064817 027516 336758 099853 406113 > 3²⁰⁶ [i]