MinT - Best Known (231−115, 231, s)-Nets in Base 4

Best Known (231−115, 231, s)-Nets in Base 4

(231−115, 231, 130)-Net over F₄ — Constructive and digital

Digital (116, 231, 130)-net over F₄, using

t-expansion [i] based on digital (105, 231, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(231−115, 231, 168)-Net over F₄ — Digital

Digital (116, 231, 168)-net over F₄, using

t-expansion [i] based on digital (115, 231, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(231−115, 231, 1931)-Net in Base 4 — Upper bound on s

There is no (116, 231, 1932)-net in base 4, because

1 times m-reduction [i] would yield (116, 230, 1932)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 022075 358168 001907 519866 112912 424663 833425 174989 458093 667802 773198 418523 984451 949364 326477 859491 950591 101481 743439 821152 009827 052824 214760 > 4²³⁰ [i]