MinT - Best Known (98, 137, s)-Nets in Base 5

Best Known (98, 137, s)-Nets in Base 5

Digital (98, 137, 400)-net over F₅, using

9 times m-reduction [i] based on digital (98, 146, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 73, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

Digital (98, 137, 1263)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³⁷, 1263, F₅, 39) (dual of [1263, 1126, 40]-code), using
- 1125 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 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, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 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, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 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, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 39 times 0, 1, 42 times 0, 1, 43 times 0, 1, 46 times 0, 1, 47 times 0, 1, 50 times 0) [i] based on linear OA(5³⁹, 40, F₅, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,5)), using
  - dual of repetition code with length 40 [i]

There is no (98, 137, 199655)-net in base 5, because

1 times m-reduction [i] would yield (98, 136, 199655)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 114798 729783 634794 300399 319491 583843 618880 860077 112238 843701 746757 866701 004022 092694 980744 183765 > 5¹³⁶ [i]