MinT - Best Known (137, 137+109, s)-Nets in Base 3

Best Known (137, 137+109, s)-Nets in Base 3

(137, 137+109, 128)-Net over F₃ — Constructive and digital

Digital (137, 246, 128)-net over F₃, using

2 times m-reduction [i] based on digital (137, 248, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 124, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(137, 137+109, 168)-Net over F₃ — Digital

Digital (137, 246, 168)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+109, 1479)-Net in Base 3 — Upper bound on s

There is no (137, 246, 1480)-net in base 3, because

1 times m-reduction [i] would yield (137, 245, 1480)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 808 233268 442188 646939 421110 987919 931621 837577 015556 106172 260583 131406 048131 525936 298092 273570 010496 328208 507639 526321 > 3²⁴⁵ [i]