MinT - Best Known (132, 132+82, s)-Nets in Base 4

Best Known (132, 132+82, s)-Nets in Base 4

(132, 132+82, 137)-Net over F₄ — Constructive and digital

Digital (132, 214, 137)-net over F₄, using

6 times m-reduction [i] based on digital (132, 220, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 59, 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 (73, 161, 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]

(132, 132+82, 358)-Net over F₄ — Digital

Digital (132, 214, 358)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(132, 132+82, 7435)-Net in Base 4 — Upper bound on s

There is no (132, 214, 7436)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 696 232523 232161 331296 457905 343631 598950 044145 643486 789229 463989 036987 258107 868065 644065 020028 101073 560127 797370 308177 403367 370680 > 4²¹⁴ [i]