MinT - Best Known (33−10, 33, s)-Nets in Base 5

Best Known (33−10, 33, s)-Nets in Base 5

(33−10, 33, 138)-Net over F₅ — Constructive and digital

Digital (23, 33, 138)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (18, 28, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 14, 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]

(33−10, 33, 583)-Net over F₅ — Digital

Digital (23, 33, 583)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³³, 583, F₅, 10) (dual of [583, 550, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(5³³, 624, F₅, 10) (dual of [624, 591, 11]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−3,−2,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(33−10, 33, 26725)-Net in Base 5 — Upper bound on s

There is no (23, 33, 26726)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 116426 693308 764628 378905 > 5³³ [i]