MinT - Best Known (57, 57+177, s)-Nets in Base 4

Best Known (57, 57+177, s)-Nets in Base 4

(57, 57+177, 66)-Net over F₄ — Constructive and digital

Digital (57, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(57, 57+177, 91)-Net over F₄ — Digital

Digital (57, 234, 91)-net over F₄, using

t-expansion [i] based on digital (50, 234, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(57, 57+177, 238)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 234, 239)-net over F₄, because

5 times m-reduction [i] would yield digital (57, 229, 239)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁹, 239, F₄, 172) (dual of [239, 10, 173]-code), but
  - residual code [i] would yield linear OA(4⁵⁷, 66, F₄, 43) (dual of [66, 9, 44]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(57, 57+177, 370)-Net in Base 4 — Upper bound on s

There is no (57, 234, 371)-net in base 4, because

1 times m-reduction [i] would yield (57, 233, 371)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 202 679929 799806 287611 029862 468426 078692 445916 611459 463947 602567 336292 133554 397306 299277 967612 737132 205604 786266 044116 415072 479583 372679 134724 > 4²³³ [i]