MinT - Best Known (121, 121+137, s)-Nets in Base 4

Best Known (121, 121+137, s)-Nets in Base 4

(121, 121+137, 130)-Net over F₄ — Constructive and digital

Digital (121, 258, 130)-net over F₄, using

t-expansion [i] based on digital (105, 258, 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]

(121, 121+137, 168)-Net over F₄ — Digital

Digital (121, 258, 168)-net over F₄, using

t-expansion [i] based on digital (115, 258, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(121, 121+137, 1588)-Net in Base 4 — Upper bound on s

There is no (121, 258, 1589)-net in base 4, because

1 times m-reduction [i] would yield (121, 257, 1589)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54845 617202 060835 597520 261463 179775 395876 288570 179909 579847 580532 782883 094527 444963 759281 301924 737141 746211 378593 182002 987245 806298 906119 348430 214369 178040 > 4²⁵⁷ [i]