MinT - Best Known (37−15, 37, s)-Nets in Base 5

Best Known (37−15, 37, s)-Nets in Base 5

(37−15, 37, 104)-Net over F₅ — Constructive and digital

Digital (22, 37, 104)-net over F₅, using

1 times m-reduction [i] based on digital (22, 38, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 19, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]

(37−15, 37, 115)-Net over F₅ — Digital

Digital (22, 37, 115)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³⁷, 115, F₅, 15) (dual of [115, 78, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5³⁷, 124, F₅, 15) (dual of [124, 87, 16]-code), using
  - the primitive BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â {−6,−5,…,8}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(37−15, 37, 3318)-Net in Base 5 — Upper bound on s

There is no (22, 37, 3319)-net in base 5, because

1 times m-reduction [i] would yield (22, 36, 3319)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 567515 884024 661672 293701 > 5³⁶ [i]