MinT - Best Known (123−45, 123, s)-Nets in Base 4

Best Known (123−45, 123, s)-Nets in Base 4

(123−45, 123, 130)-Net over F₄ — Constructive and digital

Digital (78, 123, 130)-net over F₄, using

21 times m-reduction [i] based on digital (78, 144, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 72, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(123−45, 123, 262)-Net over F₄ — Digital

Digital (78, 123, 262)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−45, 123, 6565)-Net in Base 4 — Upper bound on s

There is no (78, 123, 6566)-net in base 4, because

1 times m-reduction [i] would yield (78, 122, 6566)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28 340545 684277 129478 895323 665078 311791 072291 133815 973759 076619 260902 285024 > 4¹²² [i]