MinT - Best Known (123, 123+39, s)-Nets in Base 4

Best Known (123, 123+39, s)-Nets in Base 4

(123, 123+39, 1032)-Net over F₄ — Constructive and digital

Digital (123, 162, 1032)-net over F₄, using

4² times duplication [i] based on digital (121, 160, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 40, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(123, 123+39, 1865)-Net over F₄ — Digital

Digital (123, 162, 1865)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+39, 333995)-Net in Base 4 — Upper bound on s

There is no (123, 162, 333996)-net in base 4, because

1 times m-reduction [i] would yield (123, 161, 333996)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 544213 083264 209633 240333 370266 504803 760714 146228 583596 713635 194859 521389 990825 962003 765988 019886 > 4¹⁶¹ [i]