MinT - Best Known (51, 51+54, s)-Nets in Base 32

Best Known (51, 51+54, s)-Nets in Base 32

(51, 51+54, 240)-Net over F₃₂ — Constructive and digital

Digital (51, 105, 240)-net over F₃₂, using

4 times m-reduction [i] based on digital (51, 109, 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, 69, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(51, 51+54, 513)-Net in Base 32 — Constructive

(51, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 times m-reduction [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+54, 645)-Net over F₃₂ — Digital

Digital (51, 105, 645)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+54, 251409)-Net in Base 32 — Upper bound on s

There is no (51, 105, 251410)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 839530 148669 695898 862063 867225 550716 463880 217883 596940 103716 720136 869633 734514 925970 213095 705867 471974 945289 910016 536946 538702 778755 203248 777820 422569 442048 > 32¹⁰⁵ [i]