MinT - Best Known (207−77, 207, s)-Nets in Base 3

Best Known (207−77, 207, s)-Nets in Base 3

(207−77, 207, 156)-Net over F₃ — Constructive and digital

Digital (130, 207, 156)-net over F₃, using

9 times m-reduction [i] based on digital (130, 216, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 108, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(207−77, 207, 237)-Net over F₃ — Digital

Digital (130, 207, 237)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−77, 207, 2861)-Net in Base 3 — Upper bound on s

There is no (130, 207, 2862)-net in base 3, because

1 times m-reduction [i] would yield (130, 206, 2862)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 193 978225 853525 213707 970789 812467 115455 829802 007991 759930 444295 611945 833500 014628 108007 830662 828677 > 3²⁰⁶ [i]