MinT - Best Known (75−26, 75, s)-Nets in Base 5

Best Known (75−26, 75, s)-Nets in Base 5

Digital (49, 75, 252)-net over F₅, using

3 times m-reduction [i] based on digital (49, 78, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 39, 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 (49, 75, 331)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁵, 331, F₅, 26) (dual of [331, 256, 27]-code), using
- 255 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 1, 1, 1, 1, 1, 1, 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, 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, 6 times 0, 1, 6 times 0, 1, 6 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, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0) [i] based on linear OA(5²⁶, 27, F₅, 26) (dual of [27, 1, 27]-code or 27-arc in PG(25,5)), using
  - dual of repetition code with length 27 [i]

There is no (49, 75, 15262)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 26476 979613 080300 855633 102276 783942 976052 773728 114425 > 5⁷⁵ [i]