MinT - Best Known (133−33, 133, s)-Nets in Base 3

Best Known (133−33, 133, s)-Nets in Base 3

(133−33, 133, 328)-Net over F₃ — Constructive and digital

Digital (100, 133, 328)-net over F₃, using

3¹ times duplication [i] based on digital (99, 132, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 33, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(133−33, 133, 641)-Net over F₃ — Digital

Digital (100, 133, 641)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³³, 641, F₃, 33) (dual of [641, 508, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³³, 728, F₃, 33) (dual of [728, 595, 34]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {339,340,…,371}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(133−33, 133, 29344)-Net in Base 3 — Upper bound on s

There is no (100, 133, 29345)-net in base 3, because

1 times m-reduction [i] would yield (100, 132, 29345)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 955 156820 550502 026172 307023 807847 223377 337790 161051 763794 845505 > 3¹³² [i]