MinT - Best Known (258−68, 258, s)-Nets in Base 4

Best Known (258−68, 258, s)-Nets in Base 4

(258−68, 258, 531)-Net over F₄ — Constructive and digital

Digital (190, 258, 531)-net over F₄, using

t-expansion [i] based on digital (179, 258, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 86, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(258−68, 258, 648)-Net in Base 4 — Constructive

(190, 258, 648)-net in base 4, using

trace code for nets [i] based on (18, 86, 216)-net in base 64, using
- 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
  - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(258−68, 258, 1822)-Net over F₄ — Digital

Digital (190, 258, 1822)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−68, 258, 167053)-Net in Base 4 — Upper bound on s

There is no (190, 258, 167054)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214561 495091 829944 476640 953586 651043 731570 082394 538182 528191 259254 732987 274651 277242 977979 625093 470975 412560 460810 656361 193363 713388 630488 241987 444761 458120 > 4²⁵⁸ [i]