MinT - Best Known (49, 49+61, s)-Nets in Base 32

Best Known (49, 49+61, s)-Nets in Base 32

(49, 49+61, 218)-Net over F₃₂ — Constructive and digital

Digital (49, 110, 218)-net over F₃₂, using

32¹ times duplication [i] based on digital (48, 109, 218)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 37, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
  2. digital (11, 72, 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]

(49, 49+61, 421)-Net over F₃₂ — Digital

Digital (49, 110, 421)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+61, 513)-Net in Base 32 — Constructive

(49, 110, 513)-net in base 32, using

32² times duplication [i] based on (47, 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]

(49, 49+61, 114308)-Net in Base 32 — Upper bound on s

There is no (49, 110, 114309)-net in base 32, because

1 times m-reduction [i] would yield (49, 109, 114309)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 177151 071640 405445 928678 033676 494856 958891 400914 877101 569420 354040 101167 983881 694203 643214 334855 029213 661550 872120 178831 290511 381211 963804 025455 332579 819494 712512 > 32¹⁰⁹ [i]