MinT - Best Known (124, 124+83, s)-Nets in Base 4

Best Known (124, 124+83, s)-Nets in Base 4

(124, 124+83, 131)-Net over F₄ — Constructive and digital

Digital (124, 207, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 51, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 156, 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]

(124, 124+83, 299)-Net over F₄ — Digital

Digital (124, 207, 299)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+83, 5665)-Net in Base 4 — Upper bound on s

There is no (124, 207, 5666)-net in base 4, because

1 times m-reduction [i] would yield (124, 206, 5666)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10644 687251 286319 714946 757849 542315 415188 468949 028589 871548 597466 200887 124637 059854 137926 678669 741524 691990 284680 425466 774976 > 4²⁰⁶ [i]