MinT - Best Known (131, 170, s)-Nets in Base 3

Best Known (131, 170, s)-Nets in Base 3

Digital (131, 170, 464)-net over F₃, using

2 times m-reduction [i] based on digital (131, 172, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 43, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

Digital (131, 170, 1043)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁰, 1043, F₃, 39) (dual of [1043, 873, 40]-code), using
- 872 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (14, 7, 4, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0) [i] based on linear OA(3³⁹, 40, F₃, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,3)), using
  - dual of repetition code with length 40 [i]

There is no (131, 170, 69493)-net in base 3, because

1 times m-reduction [i] would yield (131, 169, 69493)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 430 082328 079921 687984 317631 464342 209094 531692 103860 507034 444483 404362 966879 141995 > 3¹⁶⁹ [i]