MinT - Best Known (123, 123+137, s)-Nets in Base 4

Best Known (123, 123+137, s)-Nets in Base 4

(123, 123+137, 130)-Net over F₄ — Constructive and digital

Digital (123, 260, 130)-net over F₄, using

t-expansion [i] based on digital (105, 260, 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]

(123, 123+137, 168)-Net over F₄ — Digital

Digital (123, 260, 168)-net over F₄, using

t-expansion [i] based on digital (115, 260, 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]

(123, 123+137, 1656)-Net in Base 4 — Upper bound on s

There is no (123, 260, 1657)-net in base 4, because

1 times m-reduction [i] would yield (123, 259, 1657)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 863455 232620 385085 736341 738388 547587 045503 624990 884770 079435 639870 010466 393745 387215 410656 320856 029423 294543 924019 608311 650657 056764 154444 501025 378984 517310 > 4²⁵⁹ [i]