MinT - Best Known (90, 90+42, s)-Nets in Base 5

Best Known (90, 90+42, s)-Nets in Base 5

Digital (90, 132, 296)-net over F₅, using

10 times m-reduction [i] based on digital (90, 142, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 71, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

Digital (90, 132, 739)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³², 739, F₅, 42) (dual of [739, 607, 43]-code), using
- 606 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (10, 4, 3, 2, 1, 1, 1, 1, 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, 1, 0, 0, 1, 0, 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, 4 times 0, 1, 5 times 0, 1, 5 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, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 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, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0) [i] based on linear OA(5⁴², 43, F₅, 42) (dual of [43, 1, 43]-code or 43-arc in PG(41,5)), using
  - dual of repetition code with length 43 [i]

There is no (90, 132, 53683)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 183 687000 696218 973517 847883 044582 870396 634673 621499 933056 723513 341161 835139 808232 780627 991165 > 5¹³² [i]