MinT - Best Known (110−43, 110, s)-Nets in Base 9

Best Known (110−43, 110, s)-Nets in Base 9

Digital (67, 110, 344)-net over F₉, using

10 times m-reduction [i] based on digital (67, 120, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 60, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

Digital (67, 110, 673)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹⁰, 673, F₉, 43) (dual of [673, 563, 44]-code), using
- 562 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) [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 (67, 110, 97347)-net in base 9, because

1 times m-reduction [i] would yield (67, 109, 97347)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 102 920647 710246 436962 636548 092248 580527 823725 781915 537536 125865 014136 915522 773853 506289 222081 827285 012217 > 9¹⁰⁹ [i]