MinT - Best Known (60−31, 60, s)-Nets in Base 81

Best Known (60−31, 60, s)-Nets in Base 81

(60−31, 60, 370)-Net over F₈₁ — Constructive and digital

Digital (29, 60, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 60, 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]

(60−31, 60, 1101)-Net over F₈₁ — Digital

Digital (29, 60, 1101)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶⁰, 1101, F₈₁, 31) (dual of [1101, 1041, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁶⁰, 1312, F₈₁, 31) (dual of [1312, 1252, 32]-code), using
  - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {1,6,11,…,151}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(60−31, 60, 2578595)-Net in Base 81 — Upper bound on s

There is no (29, 60, 2578596)-net in base 81, because

1 times m-reduction [i] would yield (29, 59, 2578596)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 39867 400717 059294 883687 445834 203517 234674 548367 753622 323860 053284 706332 144794 993807 522889 203741 851911 475833 707201 > 81⁵⁹ [i]