MinT - Best Known (233−121, 233, s)-Nets in Base 4

Best Known (233−121, 233, s)-Nets in Base 4

(233−121, 233, 130)-Net over F₄ — Constructive and digital

Digital (112, 233, 130)-net over F₄, using

t-expansion [i] based on digital (105, 233, 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]

(233−121, 233, 165)-Net over F₄ — Digital

Digital (112, 233, 165)-net over F₄, using

t-expansion [i] based on digital (109, 233, 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]

(233−121, 233, 1596)-Net in Base 4 — Upper bound on s

There is no (112, 233, 1597)-net in base 4, because

1 times m-reduction [i] would yield (112, 232, 1597)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 197278 255668 011428 179514 635768 291321 012517 884909 038045 098329 930565 416727 067121 888171 338700 389945 863035 439529 157678 700362 806801 676145 837665 > 4²³² [i]