MinT - Best Known (106−56, 106, s)-Nets in Base 32

Best Known (106−56, 106, s)-Nets in Base 32

(106−56, 106, 240)-Net over F₃₂ — Constructive and digital

Digital (50, 106, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 39, 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, 67, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(106−56, 106, 513)-Net in Base 32 — Constructive

(50, 106, 513)-net in base 32, using

t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 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]

(106−56, 106, 548)-Net over F₃₂ — Digital

Digital (50, 106, 548)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−56, 106, 181834)-Net in Base 32 — Upper bound on s

There is no (50, 106, 181835)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3514 950660 759978 984691 353063 857951 897717 097807 330957 353431 276444 632183 696440 997778 352911 536340 887994 361454 557504 062715 856342 530123 078074 802473 841317 836983 178000 > 32¹⁰⁶ [i]