MinT - Best Known (244−87, 244, s)-Nets in Base 4

Best Known (244−87, 244, s)-Nets in Base 4

(244−87, 244, 163)-Net over F₄ — Constructive and digital

Digital (157, 244, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 58, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (99, 186, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 93, 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]

(244−87, 244, 208)-Net in Base 4 — Constructive

(157, 244, 208)-net in base 4, using

2 times m-reduction [i] based on (157, 246, 208)-net in base 4, using
- trace code for nets [i] based on (34, 123, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
    - base change [i] based on digital (9, 100, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(244−87, 244, 517)-Net over F₄ — Digital

Digital (157, 244, 517)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−87, 244, 14177)-Net in Base 4 — Upper bound on s

There is no (157, 244, 14178)-net in base 4, because

1 times m-reduction [i] would yield (157, 243, 14178)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 962519 364004 695355 986431 447420 798793 305317 879578 452922 848668 333619 852932 868504 100457 853323 795410 376871 689220 414573 502124 094827 714478 080902 909600 > 4²⁴³ [i]