MinT - Best Known (80−25, 80, s)-Nets in Base 5

Best Known (80−25, 80, s)-Nets in Base 5

(80−25, 80, 252)-Net over F₅ — Constructive and digital

Digital (55, 80, 252)-net over F₅, using

10 times m-reduction [i] based on digital (55, 90, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 45, 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]

(80−25, 80, 578)-Net over F₅ — Digital

Digital (55, 80, 578)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁰, 578, F₅, 25) (dual of [578, 498, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁰, 624, F₅, 25) (dual of [624, 544, 26]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(80−25, 80, 52818)-Net in Base 5 — Upper bound on s

There is no (55, 80, 52819)-net in base 5, because

1 times m-reduction [i] would yield (55, 79, 52819)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 16 546402 574460 052090 852217 891519 912718 655434 036503 357009 > 5⁷⁹ [i]