MinT - Best Known (172−131, 172, s)-Nets in Base 4

Best Known (172−131, 172, s)-Nets in Base 4

(172−131, 172, 56)-Net over F₄ — Constructive and digital

Digital (41, 172, 56)-net over F₄, using

t-expansion [i] based on digital (33, 172, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(172−131, 172, 75)-Net over F₄ — Digital

Digital (41, 172, 75)-net over F₄, using

t-expansion [i] based on digital (40, 172, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(172−131, 172, 172)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 172, 173)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 169, 173)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁹, 173, F₄, 128) (dual of [173, 4, 129]-code), but
  - Griesmer bound [i]

(172−131, 172, 269)-Net in Base 4 — Upper bound on s

There is no (41, 172, 270)-net in base 4, because

1 times m-reduction [i] would yield (41, 171, 270)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 458853 967664 992397 115357 075974 479041 294594 568332 983125 180951 841004 148830 075661 253774 935452 640810 020052 > 4¹⁷¹ [i]