MinT - Best Known (53−17, 53, s)-Nets in Base 5

Best Known (53−17, 53, s)-Nets in Base 5

(53−17, 53, 208)-Net over F₅ — Constructive and digital

Digital (36, 53, 208)-net over F₅, using

1 times m-reduction [i] based on digital (36, 54, 208)-net over F₅, using
- trace code for nets [i] based on digital (9, 27, 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]

(53−17, 53, 416)-Net over F₅ — Digital

Digital (36, 53, 416)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵³, 416, F₅, 17) (dual of [416, 363, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁵³, 624, F₅, 17) (dual of [624, 571, 18]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(53−17, 53, 32874)-Net in Base 5 — Upper bound on s

There is no (36, 53, 32875)-net in base 5, because

1 times m-reduction [i] would yield (36, 52, 32875)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 220581 283889 165447 753310 110654 936801 > 5⁵² [i]