MinT - Best Known (109−33, 109, s)-Nets in Base 5

Best Known (109−33, 109, s)-Nets in Base 5

Digital (76, 109, 296)-net over F₅, using

5 times m-reduction [i] based on digital (76, 114, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 57, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

Digital (76, 109, 785)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁹, 785, F₅, 33) (dual of [785, 676, 34]-code), using
- 675 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 6 times 0, 1, 7 times 0, 1, 7 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, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 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, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0) [i] based on linear OA(5³³, 34, F₅, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,5)), using
  - dual of repetition code with length 34 [i]

There is no (76, 109, 88812)-net in base 5, because

1 times m-reduction [i] would yield (76, 108, 88812)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3081 671477 973773 514407 226648 252623 261739 323833 315366 752374 679492 651875 436545 > 5¹⁰⁸ [i]