MinT - Best Known (212−152, 212, s)-Nets in Base 4

Best Known (212−152, 212, s)-Nets in Base 4

(212−152, 212, 66)-Net over F₄ — Constructive and digital

Digital (60, 212, 66)-net over F₄, using

t-expansion [i] based on digital (49, 212, 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]

(212−152, 212, 91)-Net over F₄ — Digital

Digital (60, 212, 91)-net over F₄, using

t-expansion [i] based on digital (50, 212, 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]

(212−152, 212, 353)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 212, 354)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹², 354, F₄, 152) (dual of [354, 142, 153]-code), but
- residual code [i] would yield OA(4⁶⁰, 201, S₄, 38), but
  - the linear programming bound shows that MÂ â‰¥ 73089 680874 254803 480484 380402 731584 606836 512208 238432 667847 363593 365796 276173 537280 / 52941 428031 033877 887534 646852 798421 725736 559559 > 4⁶⁰ [i]

(212−152, 212, 404)-Net in Base 4 — Upper bound on s

There is no (60, 212, 405)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 923594 628690 629367 916849 455127 156559 533267 153000 160982 939097 539306 188410 331375 968701 854692 201611 448292 410749 105690 311157 257988 > 4²¹² [i]