MinT - Best Known (70, 70+41, s)-Nets in Base 9

Best Known (70, 70+41, s)-Nets in Base 9

Digital (70, 111, 448)-net over F₉, using

3 times m-reduction [i] based on digital (70, 114, 448)-net over F₉, using
- trace code for nets [i] based on digital (13, 57, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

Digital (70, 111, 897)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹¹, 897, F₉, 41) (dual of [897, 786, 42]-code), using
- 785 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 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, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 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, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 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, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 43 times 0, 1, 45 times 0) [i] based on linear OA(9⁴¹, 42, F₉, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,9)), using
  - dual of repetition code with length 42 [i]

There is no (70, 111, 183875)-net in base 9, because

1 times m-reduction [i] would yield (70, 110, 183875)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 926 180566 479447 282442 352554 816113 345984 562440 985650 228650 935005 052356 899312 836253 561168 579803 136803 493601 > 9¹¹⁰ [i]