MinT - Best Known (235−89, 235, s)-Nets in Base 4

Best Known (235−89, 235, s)-Nets in Base 4

(235−89, 235, 139)-Net over F₄ — Constructive and digital

Digital (146, 235, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 45, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (101, 190, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 95, 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]

(235−89, 235, 146)-Net in Base 4 — Constructive

(146, 235, 146)-net in base 4, using

3 times m-reduction [i] based on (146, 238, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 73, 42)-net in base 4, using
    - net from sequence [i] based on (27, 41)-sequence in base 4, using
      - base expansion [i] based on digital (54, 41)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 165, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(235−89, 235, 405)-Net over F₄ — Digital

Digital (146, 235, 405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−89, 235, 9119)-Net in Base 4 — Upper bound on s

There is no (146, 235, 9120)-net in base 4, because

1 times m-reduction [i] would yield (146, 234, 9120)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 765 639313 183608 289868 639781 587861 272679 417380 560813 586425 083542 012281 975335 580789 742893 603312 632246 600317 890822 121710 786709 923094 641161 645452 > 4²³⁴ [i]