MinT - Best Known (106−32, 106, s)-Nets in Base 5

Best Known (106−32, 106, s)-Nets in Base 5

Digital (74, 106, 296)-net over F₅, using

4 times m-reduction [i] based on digital (74, 110, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 55, 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 (74, 106, 778)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁶, 778, F₅, 32) (dual of [778, 672, 33]-code), using
- 671 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 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, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 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, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 35 times 0, 1, 37 times 0) [i] based on linear OA(5³², 33, F₅, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,5)), using
  - dual of repetition code with length 33 [i]

There is no (74, 106, 72626)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 123 286186 525731 720545 230004 298918 652996 952940 497670 202828 331051 942201 594625 > 5¹⁰⁶ [i]