MinT - Best Known (254−184, 254, s)-Nets in Base 4

Best Known (254−184, 254, s)-Nets in Base 4

(254−184, 254, 66)-Net over F₄ — Constructive and digital

Digital (70, 254, 66)-net over F₄, using

t-expansion [i] based on digital (49, 254, 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]

(254−184, 254, 105)-Net over F₄ — Digital

Digital (70, 254, 105)-net over F₄, using

net from sequence [i] based on digital (70, 104)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
  - Drinfeld modules of rank 1 [i]

(254−184, 254, 375)-Net over F₄ — Upper bound on s (digital)

There is no digital (70, 254, 376)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 376, F₄, 184) (dual of [376, 122, 185]-code), but
- residual code [i] would yield OA(4⁷⁰, 191, S₄, 46), but
  - the linear programming bound shows that MÂ â‰¥ 30669 687295 767807 163249 547418 721993 874758 654208 012151 612633 672044 408069 277192 616104 296206 434138 576650 240000 / 20720 048409 770029 043765 568578 931955 503324 140138 849174 037831 490849 > 4⁷⁰ [i]

(254−184, 254, 464)-Net in Base 4 — Upper bound on s

There is no (70, 254, 465)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 960 976936 397345 410183 317013 616966 741198 167742 393957 016871 019354 379881 135959 422965 808628 581814 626370 637302 118770 531852 809116 924805 718072 141663 254292 434400 > 4²⁵⁴ [i]