MinT - Best Known (238−167, 238, s)-Nets in Base 4

Best Known (238−167, 238, s)-Nets in Base 4

(238−167, 238, 66)-Net over F₄ — Constructive and digital

Digital (71, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(238−167, 238, 105)-Net over F₄ — Digital

Digital (71, 238, 105)-net over F₄, using

t-expansion [i] based on digital (70, 238, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(238−167, 238, 487)-Net in Base 4 — Upper bound on s

There is no (71, 238, 488)-net in base 4, because

1 times m-reduction [i] would yield (71, 237, 488)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49113 911373 540129 712661 239095 065425 350985 126216 483542 549551 117930 178595 245884 539986 603040 994350 695910 406363 670386 978044 784637 666689 796780 988270 > 4²³⁷ [i]