MinT - Best Known (66, 66+168, s)-Nets in Base 4

Best Known (66, 66+168, s)-Nets in Base 4

(66, 66+168, 66)-Net over F₄ — Constructive and digital

Digital (66, 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]

(66, 66+168, 99)-Net over F₄ — Digital

Digital (66, 234, 99)-net over F₄, using

t-expansion [i] based on digital (61, 234, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(66, 66+168, 382)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 234, 383)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁴, 383, F₄, 168) (dual of [383, 149, 169]-code), but
- residual code [i] would yield OA(4⁶⁶, 214, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 113 026714 592523 165071 661888 079481 443471 353704 422593 024055 302731 631761 970405 969474 549718 427811 971072 / 20291 149710 796120 930866 391420 785164 794765 888342 457078 864235 > 4⁶⁶ [i]

(66, 66+168, 442)-Net in Base 4 — Upper bound on s

There is no (66, 234, 443)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 852 565495 179782 724696 687158 024049 882077 823020 877327 468178 683336 625188 739621 931329 782968 273419 077535 228018 291578 530983 816208 167828 914555 445660 > 4²³⁴ [i]