MinT - Best Known (257−127, 257, s)-Nets in Base 4

Best Known (257−127, 257, s)-Nets in Base 4

(257−127, 257, 130)-Net over F₄ — Constructive and digital

Digital (130, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 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]

(257−127, 257, 186)-Net over F₄ — Digital

Digital (130, 257, 186)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−127, 257, 2213)-Net in Base 4 — Upper bound on s

There is no (130, 257, 2214)-net in base 4, because

1 times m-reduction [i] would yield (130, 256, 2214)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13719 431583 492515 698518 659313 804536 555752 898397 698056 479471 183086 417182 026513 525103 069481 791622 095797 565469 403065 204481 230045 809724 664020 988814 891601 227840 > 4²⁵⁶ [i]