MinT - Best Known (57, 57+148, s)-Nets in Base 4

Best Known (57, 57+148, s)-Nets in Base 4

(57, 57+148, 66)-Net over F₄ — Constructive and digital

Digital (57, 205, 66)-net over F₄, using

t-expansion [i] based on digital (49, 205, 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+148, 91)-Net over F₄ — Digital

Digital (57, 205, 91)-net over F₄, using

t-expansion [i] based on digital (50, 205, 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+148, 325)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 205, 326)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁵, 326, F₄, 148) (dual of [326, 121, 149]-code), but
- residual code [i] would yield OA(4⁵⁷, 177, S₄, 37), but
  - the linear programming bound shows that MÂ â‰¥ 7 916017 728459 289141 982805 636437 938856 538720 259539 351050 169416 698004 574623 120920 412160 / 363 505843 153703 749393 199977 171265 253621 464016 748729 > 4⁵⁷ [i]

(57, 57+148, 382)-Net in Base 4 — Upper bound on s

There is no (57, 205, 383)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3065 816714 966628 928320 353046 266430 866146 142832 156082 093590 275176 192479 593970 899453 466431 450607 003653 072426 479633 789464 249109 > 4²⁰⁵ [i]