MinT - Best Known (123, 123+59, s)-Nets in Base 3

Best Known (123, 123+59, s)-Nets in Base 3

(123, 123+59, 162)-Net over F₃ — Constructive and digital

Digital (123, 182, 162)-net over F₃, using

trace code for nets [i] based on digital (32, 91, 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]

(123, 123+59, 321)-Net over F₃ — Digital

Digital (123, 182, 321)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+59, 5517)-Net in Base 3 — Upper bound on s

There is no (123, 182, 5518)-net in base 3, because

1 times m-reduction [i] would yield (123, 181, 5518)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 228 951749 563502 603064 979468 168852 682763 407889 329550 341529 438186 802647 894959 064529 451397 > 3¹⁸¹ [i]