MinT - Best Known (252−106, 252, s)-Nets in Base 4

Best Known (252−106, 252, s)-Nets in Base 4

(252−106, 252, 137)-Net over F₄ — Constructive and digital

Digital (146, 252, 137)-net over F₄, using

t-expansion [i] based on digital (145, 252, 137)-net over F₄, using
- 7 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 72, 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, 187, 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]

(252−106, 252, 305)-Net over F₄ — Digital

Digital (146, 252, 305)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−106, 252, 4960)-Net in Base 4 — Upper bound on s

There is no (146, 252, 4961)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 810262 901356 594228 395070 698421 049830 723937 400836 894119 034480 466404 169892 107471 784602 189476 100781 763354 928277 573451 476724 064731 287548 595713 556666 101120 > 4²⁵² [i]