MinT - Best Known (77−17, 77, s)-Nets in Base 3

Best Known (77−17, 77, s)-Nets in Base 3

(77−17, 77, 464)-Net over F₃ — Constructive and digital

Digital (60, 77, 464)-net over F₃, using

3¹ times duplication [i] based on digital (59, 76, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 19, 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]

(77−17, 77, 826)-Net over F₃ — Digital

Digital (60, 77, 826)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁷, 826, F₃, 17) (dual of [826, 749, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁷, 1093, F₃, 17) (dual of [1093, 1016, 18]-code), using
  - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,3,5,…,33}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(77−17, 77, 64159)-Net in Base 3 — Upper bound on s

There is no (60, 77, 64160)-net in base 3, because

1 times m-reduction [i] would yield (60, 76, 64160)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 824905 483469 154079 347030 855136 712705 > 3⁷⁶ [i]