MinT - Best Known (104−34, 104, s)-Nets in Base 5

Best Known (104−34, 104, s)-Nets in Base 5

Digital (70, 104, 252)-net over F₅, using

16 times m-reduction [i] based on digital (70, 120, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 60, 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 (70, 104, 542)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁴, 542, F₅, 34) (dual of [542, 438, 35]-code), using
- 437 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 0, 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, 1, 0, 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, 4 times 0, 1, 5 times 0, 1, 6 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, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 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) [i] based on linear OA(5³⁴, 35, F₅, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,5)), using
  - dual of repetition code with length 35 [i]

There is no (70, 104, 33868)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4 932229 700026 060291 964976 458262 535157 282029 645359 445025 324264 292171 349809 > 5¹⁰⁴ [i]