MinT - Best Known (84, 84+25, s)-Nets in Base 5

Best Known (84, 84+25, s)-Nets in Base 5

Digital (84, 109, 400)-net over F₅, using

9 times m-reduction [i] based on digital (84, 118, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 59, 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 (84, 109, 3690)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁹, 3690, F₅, 25) (dual of [3690, 3581, 26]-code), using
- 557 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 52 times 0, 1, 97 times 0, 1, 154 times 0, 1, 205 times 0) [i] based on linear OA(5¹⁰⁰, 3124, F₅, 25) (dual of [3124, 3024, 26]-code), using
  - 1 times truncation [i] based on linear OA(5¹⁰¹, 3125, F₅, 26) (dual of [3125, 3024, 27]-code), using
    - an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

There is no (84, 109, 2582439)-net in base 5, because

1 times m-reduction [i] would yield (84, 108, 2582439)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3081 491308 104376 478045 401463 163136 116848 600308 656709 146529 873785 562030 181073 > 5¹⁰⁸ [i]