MinT - Best Known (80−21, 80, s)-Nets in Base 5

Best Known (80−21, 80, s)-Nets in Base 5

(80−21, 80, 296)-Net over F₅ — Constructive and digital

Digital (59, 80, 296)-net over F₅, using

trace code for nets [i] based on digital (19, 40, 148)-net over F₂₅, using
- net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
    - a function field by GarcÃa and Quoos [i]

(80−21, 80, 1562)-Net over F₅ — Digital

Digital (59, 80, 1562)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁰, 1562, F₅, 21) (dual of [1562, 1482, 22]-code), using
- the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {−3,466,935,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(80−21, 80, 376505)-Net in Base 5 — Upper bound on s

There is no (59, 80, 376506)-net in base 5, because

1 times m-reduction [i] would yield (59, 79, 376506)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 16 543852 074477 396248 868672 846042 892726 485101 310937 097025 > 5⁷⁹ [i]