MinT - Best Known (49−15, 49, s)-Nets in Base 5

Best Known (49−15, 49, s)-Nets in Base 5

(49−15, 49, 208)-Net over F₅ — Constructive and digital

Digital (34, 49, 208)-net over F₅, using

1 times m-reduction [i] based on digital (34, 50, 208)-net over F₅, using
- trace code for nets [i] based on digital (9, 25, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]

(49−15, 49, 531)-Net over F₅ — Digital

Digital (34, 49, 531)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴⁹, 531, F₅, 15) (dual of [531, 482, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁴⁹, 624, F₅, 15) (dual of [624, 575, 16]-code), using
  - the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−6,−5,…,8}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(49−15, 49, 52451)-Net in Base 5 — Upper bound on s

There is no (34, 49, 52452)-net in base 5, because

1 times m-reduction [i] would yield (34, 48, 52452)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3553 031465 218984 478893 711104 612625 > 5⁴⁸ [i]