MinT - Best Known (85−27, 85, s)-Nets in Base 5

Best Known (85−27, 85, s)-Nets in Base 5

(85−27, 85, 252)-Net over F₅ — Constructive and digital

Digital (58, 85, 252)-net over F₅, using

11 times m-reduction [i] based on digital (58, 96, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 48, 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]

(85−27, 85, 552)-Net over F₅ — Digital

Digital (58, 85, 552)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁵, 552, F₅, 27) (dual of [552, 467, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁵, 624, F₅, 27) (dual of [624, 539, 28]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−6,−5,…,20}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(85−27, 85, 46524)-Net in Base 5 — Upper bound on s

There is no (58, 85, 46525)-net in base 5, because

1 times m-reduction [i] would yield (58, 84, 46525)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 51706 463192 097798 545357 828068 553652 879663 879846 212786 363781 > 5⁸⁴ [i]