MinT - Best Known (239−135, 239, s)-Nets in Base 4

Best Known (239−135, 239, s)-Nets in Base 4

(239−135, 239, 104)-Net over F₄ — Constructive and digital

Digital (104, 239, 104)-net over F₄, using

t-expansion [i] based on digital (73, 239, 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]

(239−135, 239, 144)-Net over F₄ — Digital

Digital (104, 239, 144)-net over F₄, using

t-expansion [i] based on digital (91, 239, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(239−135, 239, 1128)-Net in Base 4 — Upper bound on s

There is no (104, 239, 1129)-net in base 4, because

1 times m-reduction [i] would yield (104, 238, 1129)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 201479 872272 714368 320892 777532 154688 857557 609247 680335 635075 154337 884199 676422 930769 867639 361093 272816 990596 804913 427901 157169 387031 813324 150760 > 4²³⁸ [i]