MinT - Best Known (65, 65+191, s)-Nets in Base 4

Best Known (65, 65+191, s)-Nets in Base 4

(65, 65+191, 66)-Net over F₄ — Constructive and digital

Digital (65, 256, 66)-net over F₄, using

t-expansion [i] based on digital (49, 256, 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]

(65, 65+191, 99)-Net over F₄ — Digital

Digital (65, 256, 99)-net over F₄, using

t-expansion [i] based on digital (61, 256, 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]

(65, 65+191, 281)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 256, 282)-net over F₄, because

3 times m-reduction [i] would yield digital (65, 253, 282)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵³, 282, F₄, 188) (dual of [282, 29, 189]-code), but
  - residual code [i] would yield OA(4⁶⁵, 93, S₄, 47), but
    - the linear programming bound shows that MÂ â‰¥ 21198 797041 584828 156035 107346 290870 848231 815238 299697 545216 / 15 452051 535358 751925 > 4⁶⁵ [i]

(65, 65+191, 423)-Net in Base 4 — Upper bound on s

There is no (65, 256, 424)-net in base 4, because

1 times m-reduction [i] would yield (65, 255, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3742 087152 136207 090196 647591 738602 868494 457052 180772 986029 377938 179713 114139 691275 603377 933105 872137 017621 587969 218843 679482 842605 904353 873967 366317 997688 > 4²⁵⁵ [i]