MinT - Best Known (67, 67+172, s)-Nets in Base 4

Best Known (67, 67+172, s)-Nets in Base 4

(67, 67+172, 66)-Net over F₄ — Constructive and digital

Digital (67, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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]

(67, 67+172, 99)-Net over F₄ — Digital

Digital (67, 239, 99)-net over F₄, using

t-expansion [i] based on digital (61, 239, 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]

(67, 67+172, 379)-Net over F₄ — Upper bound on s (digital)

There is no digital (67, 239, 380)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁹, 380, F₄, 172) (dual of [380, 141, 173]-code), but
- residual code [i] would yield OA(4⁶⁷, 207, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 28014 561764 993307 241664 052911 871857 702310 880709 902206 525389 902012 241710 106857 807269 754280 270102 528000 / 1 234731 917948 789567 037359 176277 481011 072922 589066 284983 746001 > 4⁶⁷ [i]

(67, 67+172, 447)-Net in Base 4 — Upper bound on s

There is no (67, 239, 448)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 816917 054301 830770 463924 275652 616589 856100 981936 577025 897966 219769 026631 028180 167092 087417 484005 976757 336259 442502 463963 263882 648874 046121 991350 > 4²³⁹ [i]