MinT - Best Known (153, 260, s)-Nets in Base 4

Best Known (153, 260, s)-Nets in Base 4

(153, 260, 138)-Net over F₄ — Constructive and digital

Digital (153, 260, 138)-net over F₄, using

t-expansion [i] based on digital (149, 260, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 184, 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]

(153, 260, 146)-Net in Base 4 — Constructive

(153, 260, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 80, 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, 180, 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]

(153, 260, 337)-Net over F₄ — Digital

Digital (153, 260, 337)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(153, 260, 5965)-Net in Base 4 — Upper bound on s

There is no (153, 260, 5966)-net in base 4, because

1 times m-reduction [i] would yield (153, 259, 5966)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 861210 889935 796641 105458 344669 115056 283048 899389 441048 612495 832000 251840 753525 750311 598067 051240 162805 333083 337630 444631 980794 923901 628026 627817 901800 418820 > 4²⁵⁹ [i]