MinT - Best Known (55−14, 55, s)-Nets in Base 5

Best Known (55−14, 55, s)-Nets in Base 5

(55−14, 55, 258)-Net over F₅ — Constructive and digital

Digital (41, 55, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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 (34, 48, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 24, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(55−14, 55, 1562)-Net over F₅ — Digital

Digital (41, 55, 1562)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁵, 1562, F₅, 14) (dual of [1562, 1507, 15]-code), using
- the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {1,42,83,…,534}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(55−14, 55, 262275)-Net in Base 5 — Upper bound on s

There is no (41, 55, 262276)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 277 558365 030910 927588 300468 920305 985425 > 5⁵⁵ [i]