MinT - Best Known (66−42, 66, s)-Nets in Base 256

Best Known (66−42, 66, s)-Nets in Base 256

(66−42, 66, 517)-Net over F₂₅₆ — Constructive and digital

Digital (24, 66, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 44, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(66−42, 66, 1025)-Net over F₂₅₆ — Digital

Digital (24, 66, 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]

(66−42, 66, 1260965)-Net in Base 256 — Upper bound on s

There is no (24, 66, 1260966)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 878 706237 741133 034122 406723 460628 718704 857925 807694 006376 383003 536226 724298 447836 443880 191411 954133 678521 343127 511172 490074 995782 965671 569133 062983 750648 331306 > 256⁶⁶ [i]