MinT - Best Known (69, 69+34, s)-Nets in Base 5

Best Known (69, 69+34, s)-Nets in Base 5

Digital (69, 103, 252)-net over F₅, using

15 times m-reduction [i] based on digital (69, 118, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 59, 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 (69, 103, 517)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰³, 517, F₅, 34) (dual of [517, 414, 35]-code), using
- 413 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) [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 (69, 103, 30807)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 986111 518835 260217 471936 302171 253145 193253 433925 995362 777724 319556 669725 > 5¹⁰³ [i]