MinT - Best Known (101−53, 101, s)-Nets in Base 32

Best Known (101−53, 101, s)-Nets in Base 32

(101−53, 101, 240)-Net over F₃₂ — Constructive and digital

Digital (48, 101, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 37, 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, 64, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(101−53, 101, 513)-Net in Base 32 — Constructive

(48, 101, 513)-net in base 32, using

t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 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]

(101−53, 101, 550)-Net over F₃₂ — Digital

Digital (48, 101, 550)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−53, 101, 209383)-Net in Base 32 — Upper bound on s

There is no (48, 101, 209384)-net in base 32, because

1 times m-reduction [i] would yield (48, 100, 209384)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 273671 562414 659998 863146 705911 337316 342097 164746 852363 089956 054692 953096 347204 399944 986298 410421 329670 386286 434259 999857 540838 952090 655125 772083 064460 > 32¹⁰⁰ [i]