MinT - Best Known (187−59, 187, s)-Nets in Base 4

Best Known (187−59, 187, s)-Nets in Base 4

(187−59, 187, 312)-Net over F₄ — Constructive and digital

Digital (128, 187, 312)-net over F₄, using

4¹ times duplication [i] based on digital (127, 186, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 62, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(187−59, 187, 640)-Net over F₄ — Digital

Digital (128, 187, 640)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−59, 187, 28256)-Net in Base 4 — Upper bound on s

There is no (128, 187, 28257)-net in base 4, because

1 times m-reduction [i] would yield (128, 186, 28257)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9619 972302 542331 826518 994919 678733 730728 712915 102517 367088 052698 311903 493685 103628 056647 429704 472057 894940 332416 > 4¹⁸⁶ [i]