MinT - Best Known (33−11, 33, s)-Nets in Base 5

Best Known (33−11, 33, s)-Nets in Base 5

(33−11, 33, 132)-Net over F₅ — Constructive and digital

Digital (22, 33, 132)-net over F₅, using

3 times m-reduction [i] based on digital (22, 36, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(33−11, 33, 313)-Net over F₅ — Digital

Digital (22, 33, 313)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(5³³, 313, F₅, 2, 11) (dual of [(313, 2), 593, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5³³, 626, F₅, 11) (dual of [626, 593, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(33−11, 33, 19369)-Net in Base 5 — Upper bound on s

There is no (22, 33, 19370)-net in base 5, because

1 times m-reduction [i] would yield (22, 32, 19370)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 23287 941462 367417 655977 > 5³² [i]