MinT - Best Known (62−37, 62, s)-Nets in Base 256

Best Known (62−37, 62, s)-Nets in Base 256

(62−37, 62, 521)-Net over F₂₅₆ — Constructive and digital

Digital (25, 62, 521)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 41, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(62−37, 62, 1025)-Net over F₂₅₆ — Digital

Digital (25, 62, 1025)-net over F₂₅₆, using

t-expansion [i] based on digital (24, 62, 1025)-net over F₂₅₆, using
- net from sequence [i] based on digital (24, 1024)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(62−37, 62, 4293848)-Net in Base 256 — Upper bound on s

There is no (25, 62, 4293849)-net in base 256, because

1 times m-reduction [i] would yield (25, 61, 4293849)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 799 170878 295720 556459 165125 550027 397439 879907 933936 889044 654285 711801 804062 098601 986716 940674 990968 732050 758420 438540 699101 545758 410365 837622 610036 > 256⁶¹ [i]