MinT - Best Known (158, 158+79, s)-Nets in Base 4

Best Known (158, 158+79, s)-Nets in Base 4

(158, 158+79, 225)-Net over F₄ — Constructive and digital

Digital (158, 237, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(118,16) in PG(236,4)) for nets [i] based on digital (40, 119, 225)-net over F₁₆, using
- net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
    - a function field by GarcÃa and Quoos [i]

(158, 158+79, 240)-Net in Base 4 — Constructive

(158, 237, 240)-net in base 4, using

7 times m-reduction [i] based on (158, 244, 240)-net in base 4, using
- trace code for nets [i] based on (36, 122, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(158, 158+79, 639)-Net over F₄ — Digital

Digital (158, 237, 639)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(158, 158+79, 22538)-Net in Base 4 — Upper bound on s

There is no (158, 237, 22539)-net in base 4, because

1 times m-reduction [i] would yield (158, 236, 22539)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12207 087065 905316 037358 376198 650012 590391 490717 960508 348524 554289 566168 641479 267167 060286 707680 370179 868889 617294 964163 462598 981670 350196 168704 > 4²³⁶ [i]