MinT - Best Known (140, 140+96, s)-Nets in Base 4

Best Known (140, 140+96, s)-Nets in Base 4

(140, 140+96, 137)-Net over F₄ — Constructive and digital

Digital (140, 236, 137)-net over F₄, using

8 times m-reduction [i] based on digital (140, 244, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 67, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 177, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(140, 140+96, 321)-Net over F₄ — Digital

Digital (140, 236, 321)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+96, 5659)-Net in Base 4 — Upper bound on s

There is no (140, 236, 5660)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12237 534945 901018 515837 635940 317845 176403 918066 748249 297147 913471 625466 324200 924229 018071 468929 984253 185596 468099 290791 262417 066665 915992 322448 > 4²³⁶ [i]