MinT - Best Known (90−29, 90, s)-Nets in Base 5

Best Known (90−29, 90, s)-Nets in Base 5

Digital (61, 90, 252)-net over F₅, using

12 times m-reduction [i] based on digital (61, 102, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 51, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

Digital (61, 90, 513)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁰, 513, F₅, 29) (dual of [513, 423, 30]-code), using
- 422 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 3, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 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, 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, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0) [i] based on linear OA(5²⁹, 30, F₅, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,5)), using
  - dual of repetition code with length 30 [i]

There is no (61, 90, 41952)-net in base 5, because

1 times m-reduction [i] would yield (61, 89, 41952)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 161 591276 739571 866895 798720 924613 600503 871085 153999 164471 924225 > 5⁸⁹ [i]