MinT - Best Known (70, 103, s)-Nets in Base 5

Best Known (70, 103, s)-Nets in Base 5

(70, 103, 262)-Net over F₅ — Constructive and digital

Digital (70, 103, 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 (53, 86, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 43, 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]

(70, 103, 599)-Net over F₅ — Digital

Digital (70, 103, 599)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰³, 599, F₅, 33) (dual of [599, 496, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁰³, 624, F₅, 33) (dual of [624, 521, 34]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(70, 103, 48563)-Net in Base 5 — Upper bound on s

There is no (70, 103, 48564)-net in base 5, because

1 times m-reduction [i] would yield (70, 102, 48564)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 197249 812629 324555 292747 713711 166541 452216 771307 209154 167080 097333 673985 > 5¹⁰² [i]