MinT - Best Known (232−121, 232, s)-Nets in Base 4

Best Known (232−121, 232, s)-Nets in Base 4

(232−121, 232, 130)-Net over F₄ — Constructive and digital

Digital (111, 232, 130)-net over F₄, using

t-expansion [i] based on digital (105, 232, 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]

(232−121, 232, 165)-Net over F₄ — Digital

Digital (111, 232, 165)-net over F₄, using

t-expansion [i] based on digital (109, 232, 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]

(232−121, 232, 1558)-Net in Base 4 — Upper bound on s

There is no (111, 232, 1559)-net in base 4, because

1 times m-reduction [i] would yield (111, 231, 1559)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 108222 442866 094101 885109 853204 539335 433778 396142 348184 797595 857966 207612 057824 800205 184438 387405 312201 503157 834088 907319 921933 883601 270080 > 4²³¹ [i]