MinT - Best Known (66−34, 66, s)-Nets in Base 81

Best Known (66−34, 66, s)-Nets in Base 81

Digital (32, 66, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 66, 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]

Digital (32, 66, 1189)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶⁶, 1189, F₈₁, 34) (dual of [1189, 1123, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁶⁶, 1313, F₈₁, 34) (dual of [1313, 1247, 35]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(81⁶⁵, 1312, F₈₁, 34) (dual of [1312, 1247, 35]-code), using
    - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {1,54,107,…,438}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

There is no (32, 66, 2302853)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 912038 701853 090385 001974 527057 126456 415615 127704 131582 562471 993937 129918 122076 904300 467590 852763 569829 623445 737480 132203 342481 > 81⁶⁶ [i]