MinT - Best Known (13, 13+9, s)-Nets in Base 5

Best Known (13, 13+9, s)-Nets in Base 5

(13, 13+9, 58)-Net over F₅ — Constructive and digital

Digital (13, 22, 58)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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 (9, 18, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 9, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [i]

(13, 13+9, 102)-Net over F₅ — Digital

Digital (13, 22, 102)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²², 102, F₅, 9) (dual of [102, 80, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(5²², 124, F₅, 9) (dual of [124, 102, 10]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(13, 13+9, 2583)-Net in Base 5 — Upper bound on s

There is no (13, 22, 2584)-net in base 5, because

1 times m-reduction [i] would yield (13, 21, 2584)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 477 395153 332225 > 5²¹ [i]