MinT - Best Known (69, 101, s)-Nets in Base 5

Best Known (69, 101, s)-Nets in Base 5

(69, 101, 262)-Net over F₅ — Constructive and digital

Digital (69, 101, 262)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (52, 84, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 42, 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]

(69, 101, 624)-Net over F₅ — Digital

Digital (69, 101, 624)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰¹, 624, F₅, 32) (dual of [624, 523, 33]-code), using
- the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−6,−5,…,25}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(69, 101, 43915)-Net in Base 5 — Upper bound on s

There is no (69, 101, 43916)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 39456 984652 057525 785908 571844 175880 890180 589308 500704 981364 041183 340545 > 5¹⁰¹ [i]