MinT - Best Known (55−20, 55, s)-Nets in Base 5

Best Known (55−20, 55, s)-Nets in Base 5

Digital (35, 55, 132)-net over F₅, using

7 times m-reduction [i] based on digital (35, 62, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 31, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

Digital (35, 55, 219)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁵, 219, F₅, 20) (dual of [219, 164, 21]-code), using
- 163 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 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, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0) [i] based on linear OA(5²⁰, 21, F₅, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,5)), using
  - dual of repetition code with length 21 [i]

There is no (35, 55, 7905)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 277 778357 115455 810261 265247 624557 569785 > 5⁵⁵ [i]