MinT - Best Known (226−79, 226, s)-Nets in Base 4

Best Known (226−79, 226, s)-Nets in Base 4

(226−79, 226, 163)-Net over F₄ — Constructive and digital

Digital (147, 226, 163)-net over F₄, using

1 times m-reduction [i] based on digital (147, 227, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 55, 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 (92, 172, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 86, 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]

(226−79, 226, 240)-Net in Base 4 — Constructive

(147, 226, 240)-net in base 4, using

trace code for nets [i] based on (34, 113, 120)-net in base 16, using
- 2 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 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]

(226−79, 226, 517)-Net over F₄ — Digital

Digital (147, 226, 517)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−79, 226, 15234)-Net in Base 4 — Upper bound on s

There is no (147, 226, 15235)-net in base 4, because

1 times m-reduction [i] would yield (147, 225, 15235)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2914 136384 060249 506296 933051 120873 429557 651457 198624 729858 516390 311645 608209 882096 434403 997312 553139 392075 619770 845369 500323 290411 221760 > 4²²⁵ [i]