MinT - Best Known (168, 168+39, s)-Nets in Base 3

Best Known (168, 168+39, s)-Nets in Base 3

(168, 168+39, 700)-Net over F₃ — Constructive and digital

Digital (168, 207, 700)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (145, 184, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 46, 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]

(168, 168+39, 3280)-Net over F₃ — Digital

Digital (168, 207, 3280)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁷, 3280, F₃, 39) (dual of [3280, 3073, 40]-code), using
- 1 times truncation [i] based on linear OA(3²⁰⁸, 3281, F₃, 40) (dual of [3281, 3073, 41]-code), using
  - the BCH-code C(I) with length 3281Â | 3¹⁶âˆ’1, defining interval IÂ =Â {729,944,1159,…,−729}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]

(168, 168+39, 590433)-Net in Base 3 — Upper bound on s

There is no (168, 207, 590434)-net in base 3, because

1 times m-reduction [i] would yield (168, 206, 590434)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 193 637578 283254 305808 438771 667623 079673 112855 912253 964584 245566 388211 343011 507007 494683 583200 189233 > 3²⁰⁶ [i]