MinT - Best Known (20, 20+13, s)-Nets in Base 9

Best Known (20, 20+13, s)-Nets in Base 9

(20, 20+13, 232)-Net over F₉ — Constructive and digital

Digital (20, 33, 232)-net over F₉, using

3 times m-reduction [i] based on digital (20, 36, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 18, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(20, 20+13, 361)-Net over F₉ — Digital

Digital (20, 33, 361)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³³, 361, F₉, 13) (dual of [361, 328, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(9³³, 364, F₉, 13) (dual of [364, 331, 14]-code), using
  - the BCH-code C(I) with length 364Â | 9³âˆ’1, defining interval IÂ =Â {97,126,155,…,81}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(20, 20+13, 45961)-Net in Base 9 — Upper bound on s

There is no (20, 33, 45962)-net in base 9, because

1 times m-reduction [i] would yield (20, 32, 45962)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3 433880 522451 869501 345158 592545 > 9³² [i]