MinT - Best Known (249−129, 249, s)-Nets in Base 4

Best Known (249−129, 249, s)-Nets in Base 4

(249−129, 249, 130)-Net over F₄ — Constructive and digital

Digital (120, 249, 130)-net over F₄, using

t-expansion [i] based on digital (105, 249, 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]

(249−129, 249, 168)-Net over F₄ — Digital

Digital (120, 249, 168)-net over F₄, using

t-expansion [i] based on digital (115, 249, 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]

(249−129, 249, 1718)-Net in Base 4 — Upper bound on s

There is no (120, 249, 1719)-net in base 4, because

1 times m-reduction [i] would yield (120, 248, 1719)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209946 573193 264731 081528 738431 185399 329707 882374 352489 836401 603303 155638 444288 222379 071751 413055 261205 439249 002912 334910 585806 530722 731771 141971 938770 > 4²⁴⁸ [i]