MinT - Best Known (72, 72+69, s)-Nets in Base 4

Best Known (72, 72+69, s)-Nets in Base 4

(72, 72+69, 73)-Net over F₄ — Constructive and digital

Digital (72, 141, 73)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 39, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 102, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(72, 72+69, 114)-Net over F₄ — Digital

Digital (72, 141, 114)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+69, 1332)-Net in Base 4 — Upper bound on s

There is no (72, 141, 1333)-net in base 4, because

1 times m-reduction [i] would yield (72, 140, 1333)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 990577 493766 760876 657771 383563 775419 514971 151760 388369 881788 642581 081699 992692 099906 > 4¹⁴⁰ [i]