MinT - Best Known (228−42, 228, s)-Nets in Base 3

Best Known (228−42, 228, s)-Nets in Base 3

(228−42, 228, 896)-Net over F₃ — Constructive and digital

Digital (186, 228, 896)-net over F₃, using

t-expansion [i] based on digital (184, 228, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 57, 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]

(228−42, 228, 3984)-Net over F₃ — Digital

Digital (186, 228, 3984)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, 3984, F₃, 42) (dual of [3984, 3756, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁸, 6575, F₃, 42) (dual of [6575, 6347, 43]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,19]) [i] based on
    1. linear OA(3²²⁵, 6562, F₃, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(3²⁰⁹, 6562, F₃, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(228−42, 228, 657086)-Net in Base 3 — Upper bound on s

There is no (186, 228, 657087)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 076463 588670 317495 222254 312865 304325 411922 531313 465018 492040 538502 410705 190762 586704 146698 129672 541741 369175 > 3²²⁸ [i]