MinT - Best Known (122, 122+59, s)-Nets in Base 3

Best Known (122, 122+59, s)-Nets in Base 3

(122, 122+59, 156)-Net over F₃ — Constructive and digital

Digital (122, 181, 156)-net over F₃, using

19 times m-reduction [i] based on digital (122, 200, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 100, 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]

(122, 122+59, 314)-Net over F₃ — Digital

Digital (122, 181, 314)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+59, 5311)-Net in Base 3 — Upper bound on s

There is no (122, 181, 5312)-net in base 3, because

1 times m-reduction [i] would yield (122, 180, 5312)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 76 393647 323702 420985 291068 770129 907500 626009 285048 223269 864766 693753 898080 143257 867649 > 3¹⁸⁰ [i]