MinT - Best Known (127, 127+30, s)-Nets in Base 3

Best Known (127, 127+30, s)-Nets in Base 3

Digital (127, 157, 688)-net over F₃, using

3 times m-reduction [i] based on digital (127, 160, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 40, 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]

Digital (127, 157, 2375)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁷, 2375, F₃, 30) (dual of [2375, 2218, 31]-code), using
- 171 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 31 times 0, 1, 40 times 0) [i] based on linear OA(3¹⁴⁰, 2187, F₃, 30) (dual of [2187, 2047, 31]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]

There is no (127, 157, 316655)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 809 177621 133471 475983 737802 181474 513519 031412 427638 553793 690396 287286 410147 > 3¹⁵⁷ [i]