MinT - Best Known (258−99, 258, s)-Nets in Base 4

Best Known (258−99, 258, s)-Nets in Base 4

(258−99, 258, 160)-Net over F₄ — Constructive and digital

Digital (159, 258, 160)-net over F₄, using

t-expansion [i] based on digital (157, 258, 160)-net over F₄, using
- 1 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 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]
    2. digital (73, 175, 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]

(258−99, 258, 421)-Net over F₄ — Digital

Digital (159, 258, 421)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−99, 258, 9120)-Net in Base 4 — Upper bound on s

There is no (159, 258, 9121)-net in base 4, because

1 times m-reduction [i] would yield (159, 257, 9121)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53784 196246 283406 177819 728992 006421 157758 455670 335879 531674 621886 624165 174680 446024 319234 505151 179140 331698 825138 455276 088647 288558 258134 096895 875405 252576 > 4²⁵⁷ [i]