MinT - Best Known (126, 126+83, s)-Nets in Base 4

Best Known (126, 126+83, s)-Nets in Base 4

(126, 126+83, 132)-Net over F₄ — Constructive and digital

Digital (126, 209, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 53, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, 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]

(126, 126+83, 311)-Net over F₄ — Digital

Digital (126, 209, 311)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+83, 6063)-Net in Base 4 — Upper bound on s

There is no (126, 209, 6064)-net in base 4, because

1 times m-reduction [i] would yield (126, 208, 6064)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169451 705538 069766 421138 379163 273384 380954 995385 500605 279220 879112 840630 911983 398677 303173 619467 805695 307018 391203 145449 192935 > 4²⁰⁸ [i]