MinT - Best Known (60−16, 60, s)-Nets in Base 5

Best Known (60−16, 60, s)-Nets in Base 5

(60−16, 60, 258)-Net over F₅ — Constructive and digital

Digital (44, 60, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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 (36, 52, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 26, 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]

(60−16, 60, 1324)-Net over F₅ — Digital

Digital (44, 60, 1324)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶⁰, 1324, F₅, 16) (dual of [1324, 1264, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶⁰, 1563, F₅, 16) (dual of [1563, 1503, 17]-code), using
  - the BCH-code C(I) with length 1563Â | 5¹⁰âˆ’1, defining interval IÂ =Â {3,211,419,…,−3}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(60−16, 60, 164395)-Net in Base 5 — Upper bound on s

There is no (44, 60, 164396)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 867372 425687 729282 804708 246988 782198 431745 > 5⁶⁰ [i]