MinT - Best Known (245−103, 245, s)-Nets in Base 4

Best Known (245−103, 245, s)-Nets in Base 4

(245−103, 245, 137)-Net over F₄ — Constructive and digital

Digital (142, 245, 137)-net over F₄, using

5 times m-reduction [i] based on digital (142, 250, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 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, 181, 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]

(245−103, 245, 298)-Net over F₄ — Digital

Digital (142, 245, 298)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−103, 245, 4983)-Net in Base 4 — Upper bound on s

There is no (142, 245, 4984)-net in base 4, because

1 times m-reduction [i] would yield (142, 244, 4984)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 802 225445 788320 125332 350832 400139 625226 632270 971638 630687 012543 921844 402064 942974 248370 275011 057538 804417 685368 822357 061830 345097 381094 326981 874483 > 4²⁴⁴ [i]