MinT - Best Known (182−61, 182, s)-Nets in Base 4

Best Known (182−61, 182, s)-Nets in Base 4

(182−61, 182, 163)-Net over F₄ — Constructive and digital

Digital (121, 182, 163)-net over F₄, using

2 times m-reduction [i] based on digital (121, 184, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 46, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (75, 138, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 69, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(182−61, 182, 240)-Net in Base 4 — Constructive

(121, 182, 240)-net in base 4, using

4² times duplication [i] based on (119, 180, 240)-net in base 4, using
- trace code for nets [i] based on (29, 90, 120)-net in base 16, using
  - base change [i] based on 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]

(182−61, 182, 497)-Net over F₄ — Digital

Digital (121, 182, 497)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(182−61, 182, 17198)-Net in Base 4 — Upper bound on s

There is no (121, 182, 17199)-net in base 4, because

1 times m-reduction [i] would yield (121, 181, 17199)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 410435 437405 235987 919239 399343 880436 346947 991441 937979 740089 798694 253067 466784 814368 284575 083791 284109 313290 > 4¹⁸¹ [i]