MinT - Best Known (218−43, 218, s)-Nets in Base 3

Best Known (218−43, 218, s)-Nets in Base 3

Digital (175, 218, 688)-net over F₃, using

6 times m-reduction [i] based on digital (175, 224, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 56, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

Digital (175, 218, 2507)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁸, 2507, F₃, 43) (dual of [2507, 2289, 44]-code), using
- 298 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0, 1, 22 times 0, 1, 27 times 0, 1, 33 times 0, 1, 40 times 0, 1, 47 times 0, 1, 53 times 0) [i] based on linear OA(3¹⁹⁷, 2188, F₃, 43) (dual of [2188, 1991, 44]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]

There is no (175, 218, 369566)-net in base 3, because

1 times m-reduction [i] would yield (175, 217, 369566)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 302365 442665 320274 326180 479987 720626 958734 318521 116463 118394 804202 866709 603623 140639 783922 594125 115189 > 3²¹⁷ [i]