MinT - Best Known (65−41, 65, s)-Nets in Base 256

Best Known (65−41, 65, s)-Nets in Base 256

(65−41, 65, 518)-Net over F₂₅₆ — Constructive and digital

Digital (24, 65, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 43, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(65−41, 65, 1025)-Net over F₂₅₆ — Digital

Digital (24, 65, 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]

(65−41, 65, 1656271)-Net in Base 256 — Upper bound on s

There is no (24, 65, 1656272)-net in base 256, because

1 times m-reduction [i] would yield (24, 64, 1656272)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 13407 951270 216893 393023 996981 922382 909423 621931 126189 077703 746925 952976 874426 412387 339250 642826 448570 542964 880811 975241 427742 344094 906717 921679 763909 612076 > 256⁶⁴ [i]