MinT - Best Known (185−32, 185, s)-Nets in Base 3

Best Known (185−32, 185, s)-Nets in Base 3

(185−32, 185, 896)-Net over F₃ — Constructive and digital

Digital (153, 185, 896)-net over F₃, using

3¹ times duplication [i] based on digital (152, 184, 896)-net over F₃, using
- t-expansion [i] based on digital (151, 184, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 46, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(185−32, 185, 5054)-Net over F₃ — Digital

Digital (153, 185, 5054)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁵, 5054, F₃, 32) (dual of [5054, 4869, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁵, 6578, F₃, 32) (dual of [6578, 6393, 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(3¹⁶⁹, 6561, F₃, 32) (dual of [6561, 6392, 33]-code), using
      - an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(185−32, 185, 1117421)-Net in Base 3 — Upper bound on s

There is no (153, 185, 1117422)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18511 099016 277219 517935 806326 247270 592032 178921 708694 278497 877966 447825 550447 137457 474465 > 3¹⁸⁵ [i]