MinT - Best Known (218−107, 218, s)-Nets in Base 4

Best Known (218−107, 218, s)-Nets in Base 4

(218−107, 218, 130)-Net over F₄ — Constructive and digital

Digital (111, 218, 130)-net over F₄, using

t-expansion [i] based on digital (105, 218, 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]

(218−107, 218, 165)-Net over F₄ — Digital

Digital (111, 218, 165)-net over F₄, using

t-expansion [i] based on digital (109, 218, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(218−107, 218, 1959)-Net in Base 4 — Upper bound on s

There is no (111, 218, 1960)-net in base 4, because

1 times m-reduction [i] would yield (111, 217, 1960)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44577 702678 881523 287557 611181 376390 104080 301613 815869 727351 549776 082985 453041 166390 384661 638168 669764 501917 156278 123302 081980 472360 > 4²¹⁷ [i]