MinT - Best Known (22, 22+25, s)-Nets in Base 81

Best Known (22, 22+25, s)-Nets in Base 81

(22, 22+25, 370)-Net over F₈₁ — Constructive and digital

Digital (22, 47, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 47, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(22, 22+25, 763)-Net over F₈₁ — Digital

Digital (22, 47, 763)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴⁷, 763, F₈₁, 25) (dual of [763, 716, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴⁷, 820, F₈₁, 25) (dual of [820, 773, 26]-code), using
  - the BCH-code C(I) with length 820Â | 81²âˆ’1, defining interval IÂ =Â {1,18,35,…,409}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(22, 22+25, 1368134)-Net in Base 81 — Upper bound on s

There is no (22, 47, 1368135)-net in base 81, because

1 times m-reduction [i] would yield (22, 46, 1368135)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 6170 379012 941481 513083 352922 169791 617585 249264 855879 607622 207120 133877 885022 937824 641601 > 81⁴⁶ [i]