MinT - Best Known (57, 57+172, s)-Nets in Base 4

Best Known (57, 57+172, s)-Nets in Base 4

(57, 57+172, 66)-Net over F₄ — Constructive and digital

Digital (57, 229, 66)-net over F₄, using

t-expansion [i] based on digital (49, 229, 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+172, 91)-Net over F₄ — Digital

Digital (57, 229, 91)-net over F₄, using

t-expansion [i] based on digital (50, 229, 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+172, 238)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 229, 239)-net over F₄, because

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+172, 371)-Net in Base 4 — Upper bound on s

There is no (57, 229, 372)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 811186 582280 564563 310636 539512 190139 719957 965158 477345 792241 016403 832642 513580 440177 451049 916713 391147 494454 436241 967459 710704 432436 578552 > 4²²⁹ [i]