MinT - Best Known (159, 159+83, s)-Nets in Base 3

Best Known (159, 159+83, s)-Nets in Base 3

(159, 159+83, 162)-Net over F₃ — Constructive and digital

Digital (159, 242, 162)-net over F₃, using

t-expansion [i] based on digital (157, 242, 162)-net over F₃, using
- 8 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 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]

(159, 159+83, 344)-Net over F₃ — Digital

Digital (159, 242, 344)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+83, 5105)-Net in Base 3 — Upper bound on s

There is no (159, 242, 5106)-net in base 3, because

1 times m-reduction [i] would yield (159, 241, 5106)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 757386 506316 300248 064756 632043 775870 748991 428313 922526 524947 932294 649064 166500 881282 798734 983074 366210 830482 652373 > 3²⁴¹ [i]