MinT - Best Known (87, 131, s)-Nets in Base 9

Best Known (87, 131, s)-Nets in Base 9

Digital (87, 131, 740)-net over F₉, using

11 times m-reduction [i] based on digital (87, 142, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 71, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

Digital (87, 131, 1726)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³¹, 1726, F₉, 44) (dual of [1726, 1595, 45]-code), using
- 1594 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 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, 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, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 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, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 55 times 0, 1, 57 times 0, 1, 61 times 0, 1, 63 times 0, 1, 68 times 0, 1, 70 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0) [i] based on linear OA(9⁴⁴, 45, F₉, 44) (dual of [45, 1, 45]-code or 45-arc in PG(43,9)), using
  - dual of repetition code with length 45 [i]

There is no (87, 131, 544294)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101340 662275 908773 784113 781060 411023 512320 799675 056152 282447 443186 237967 451666 065689 846230 027652 991325 395189 272963 315522 798305 > 9¹³¹ [i]