MinT - Best Known (231−99, 231, s)-Nets in Base 4

Best Known (231−99, 231, s)-Nets in Base 4

(231−99, 231, 131)-Net over F₄ — Constructive and digital

Digital (132, 231, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 59, 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, 172, 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]

(231−99, 231, 266)-Net over F₄ — Digital

Digital (132, 231, 266)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−99, 231, 4227)-Net in Base 4 — Upper bound on s

There is no (132, 231, 4228)-net in base 4, because

1 times m-reduction [i] would yield (132, 230, 4228)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 997581 805030 931477 539917 230805 179777 644869 880725 534363 319855 056316 774852 283759 009433 308696 504937 722562 968097 812540 407124 009347 677548 359016 > 4²³⁰ [i]