MinT - Best Known (178−56, 178, s)-Nets in Base 3

Best Known (178−56, 178, s)-Nets in Base 3

(178−56, 178, 162)-Net over F₃ — Constructive and digital

Digital (122, 178, 162)-net over F₃, using

2 times m-reduction [i] based on digital (122, 180, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 90, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(178−56, 178, 345)-Net over F₃ — Digital

Digital (122, 178, 345)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−56, 178, 6069)-Net in Base 3 — Upper bound on s

There is no (122, 178, 6070)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 488805 335373 335716 670119 089118 570263 035976 434009 413350 111923 795344 422377 037544 039417 > 3¹⁷⁸ [i]