MinT - Best Known (32−10, 32, s)-Nets in Base 5

Best Known (32−10, 32, s)-Nets in Base 5

(32−10, 32, 132)-Net over F₅ — Constructive and digital

Digital (22, 32, 132)-net over F₅, using

4 times m-reduction [i] based on digital (22, 36, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(32−10, 32, 476)-Net over F₅ — Digital

Digital (22, 32, 476)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³², 476, F₅, 10) (dual of [476, 444, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(5³², 624, F₅, 10) (dual of [624, 592, 11]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(32−10, 32, 19369)-Net in Base 5 — Upper bound on s

There is no (22, 32, 19370)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 23287 941462 367417 655977 > 5³² [i]