MinT - Best Known (51, 51+59, s)-Nets in Base 32

Best Known (51, 51+59, s)-Nets in Base 32

(51, 51+59, 240)-Net over F₃₂ — Constructive and digital

Digital (51, 110, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 40, 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]
2. digital (11, 70, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(51, 51+59, 513)-Net in Base 32 — Constructive

(51, 110, 513)-net in base 32, using

32² times duplication [i] based on (49, 108, 513)-net in base 32, using
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(51, 51+59, 514)-Net over F₃₂ — Digital

Digital (51, 110, 514)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+59, 171005)-Net in Base 32 — Upper bound on s

There is no (51, 110, 171006)-net in base 32, because

1 times m-reduction [i] would yield (51, 109, 171006)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 180599 121063 646507 907453 746948 542928 060325 021862 007662 558911 494786 686243 155190 562729 761116 567777 112245 854392 198542 946384 734762 192278 406296 475296 638953 810121 500497 > 32¹⁰⁹ [i]