MinT - Best Known (89, 132, s)-Nets in Base 9

Best Known (89, 132, s)-Nets in Base 9

Digital (89, 132, 740)-net over F₉, using

14 times m-reduction [i] based on digital (89, 146, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 73, 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 (89, 132, 2081)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³², 2081, F₉, 43) (dual of [2081, 1949, 44]-code), using
- 1948 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 1, 1, 1, 1, 1, 1, 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, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0, 1, 41 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 61 times 0, 1, 63 times 0, 1, 67 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 93 times 0, 1, 98 times 0, 1, 103 times 0) [i] based on linear OA(9⁴³, 44, F₉, 43) (dual of [44, 1, 44]-code or 44-arc in PG(42,9)), using
  - dual of repetition code with length 44 [i]

There is no (89, 132, 972866)-net in base 9, because

1 times m-reduction [i] would yield (89, 131, 972866)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101339 255582 848767 596316 195409 228974 847580 892116 681999 410497 601235 089907 936070 197112 543416 659782 553168 825588 315044 064117 799121 > 9¹³¹ [i]