MinT - Best Known (82, 82+159, s)-Nets in Base 4

Best Known (82, 82+159, s)-Nets in Base 4

(82, 82+159, 104)-Net over F₄ — Constructive and digital

Digital (82, 241, 104)-net over F₄, using

t-expansion [i] based on digital (73, 241, 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]

(82, 82+159, 129)-Net over F₄ — Digital

Digital (82, 241, 129)-net over F₄, using

t-expansion [i] based on digital (81, 241, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(82, 82+159, 616)-Net in Base 4 — Upper bound on s

There is no (82, 241, 617)-net in base 4, because

1 times m-reduction [i] would yield (82, 240, 617)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 249689 448053 871474 786853 477530 380500 103172 949618 496500 526151 584838 910158 760980 760078 307214 444737 774233 927547 226268 477807 390986 281917 337755 660384 > 4²⁴⁰ [i]