MinT - Best Known (79−15, 79, s)-Nets in Base 3

Best Known (79−15, 79, s)-Nets in Base 3

Digital (64, 79, 600)-net over F₃, using

1 times m-reduction [i] based on digital (64, 80, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 20, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

Digital (64, 79, 2054)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁹, 2054, F₃, 15) (dual of [2054, 1975, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁹, 3280, F₃, 15) (dual of [3280, 3201, 16]-code), using
  - 1 times truncation [i] based on linear OA(3⁸⁰, 3281, F₃, 16) (dual of [3281, 3201, 17]-code), using
    - the BCH-code C(I) with length 3281Â | 3¹⁶âˆ’1, defining interval IÂ =Â {729,1288,1847,…,−729}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

There is no (64, 79, 350248)-net in base 3, because

1 times m-reduction [i] would yield (64, 78, 350248)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 423491 831503 430781 449210 666572 716257 > 3⁷⁸ [i]