MinT - Best Known (258−110, 258, s)-Nets in Base 4

Best Known (258−110, 258, s)-Nets in Base 4

(258−110, 258, 137)-Net over F₄ — Constructive and digital

Digital (148, 258, 137)-net over F₄, using

t-expansion [i] based on digital (145, 258, 137)-net over F₄, using
- 1 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 72, 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, 187, 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]

(258−110, 258, 298)-Net over F₄ — Digital

Digital (148, 258, 298)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−110, 258, 4699)-Net in Base 4 — Upper bound on s

There is no (148, 258, 4700)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214698 328350 196883 419781 480563 777987 312336 047946 701504 757858 914371 964138 407414 848628 821143 010597 402538 928500 855796 559471 902555 829857 060887 623500 871934 987908 > 4²⁵⁸ [i]