MinT - Best Known (64, 253, s)-Nets in Base 4

Best Known (64, 253, s)-Nets in Base 4

(64, 253, 66)-Net over F₄ — Constructive and digital

Digital (64, 253, 66)-net over F₄, using

t-expansion [i] based on digital (49, 253, 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]

(64, 253, 99)-Net over F₄ — Digital

Digital (64, 253, 99)-net over F₄, using

t-expansion [i] based on digital (61, 253, 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]

(64, 253, 270)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 253, 271)-net over F₄, because

1 times m-reduction [i] would yield digital (64, 252, 271)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵², 271, F₄, 188) (dual of [271, 19, 189]-code), but
  - residual code [i] would yield OA(4⁶⁴, 82, S₄, 47), but
    - the linear programming bound shows that MÂ â‰¥ 42 906360 577844 236924 181238 144026 607595 077872 123904 / 112185 734375 > 4⁶⁴ [i]

(64, 253, 416)-Net in Base 4 — Upper bound on s

There is no (64, 253, 417)-net in base 4, because

1 times m-reduction [i] would yield (64, 252, 417)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53 887150 799342 517910 420379 902601 184060 545786 935535 486973 287432 562360 536686 057441 471765 562692 948649 573875 505600 603954 046312 944168 246831 559958 573287 868384 > 4²⁵² [i]