MinT - Best Known (100, 100+39, s)-Nets in Base 5

Best Known (100, 100+39, s)-Nets in Base 5

Digital (100, 139, 408)-net over F₅, using

1 times m-reduction [i] based on digital (100, 140, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 70, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

Digital (100, 139, 1373)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³⁹, 1373, F₅, 39) (dual of [1373, 1234, 40]-code), using
- 1233 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 6 times 0, 1, 6 times 0, 1, 6 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, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 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, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 39 times 0, 1, 42 times 0, 1, 43 times 0, 1, 46 times 0, 1, 47 times 0, 1, 50 times 0, 1, 52 times 0, 1, 54 times 0) [i] based on linear OA(5³⁹, 40, F₅, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,5)), using
  - dual of repetition code with length 40 [i]

There is no (100, 139, 236516)-net in base 5, because

1 times m-reduction [i] would yield (100, 138, 236516)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 869953 673168 380375 442177 643133 912052 238721 341529 054867 129986 454514 909280 397807 520100 855264 276625 > 5¹³⁸ [i]