MinT - Best Known (251−89, 251, s)-Nets in Base 4

Best Known (251−89, 251, s)-Nets in Base 4

(251−89, 251, 200)-Net over F₄ — Constructive and digital

Digital (162, 251, 200)-net over F₄, using

t-expansion [i] based on digital (161, 251, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−89, 251, 240)-Net in Base 4 — Constructive

(162, 251, 240)-net in base 4, using

4¹ times duplication [i] based on (161, 250, 240)-net in base 4, using
- trace code for nets [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]

(251−89, 251, 540)-Net over F₄ — Digital

Digital (162, 251, 540)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−89, 251, 15120)-Net in Base 4 — Upper bound on s

There is no (162, 251, 15121)-net in base 4, because

1 times m-reduction [i] would yield (162, 250, 15121)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 280102 271076 479140 361770 710345 204395 915215 465445 807683 291417 294578 186073 001890 908611 938144 268192 660539 844263 237417 001927 326056 940805 604359 517618 590176 > 4²⁵⁰ [i]