MinT - Best Known (89, 125, s)-Nets in Base 4

Best Known (89, 125, s)-Nets in Base 4

Digital (89, 125, 384)-net over F₄, using

1 times m-reduction [i] based on digital (89, 126, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 42, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

Digital (89, 125, 673)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁵, 673, F₄, 36) (dual of [673, 548, 37]-code), using
- 547 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (10, 5, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 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, 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, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 6 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, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0) [i] based on linear OA(4³⁶, 37, F₄, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,4)), using
  - dual of repetition code with length 37 [i]

There is no (89, 125, 38179)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1809 886657 872668 812230 868526 053115 770278 417583 926039 288165 305060 069056 824118 > 4¹²⁵ [i]