MinT - Best Known (67−42, 67, s)-Nets in Base 256

Best Known (67−42, 67, s)-Nets in Base 256

(67−42, 67, 518)-Net over F₂₅₆ — Constructive and digital

Digital (25, 67, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-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 (see above)

(67−42, 67, 1025)-Net over F₂₅₆ — Digital

Digital (25, 67, 1025)-net over F₂₅₆, using

t-expansion [i] based on digital (24, 67, 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]

(67−42, 67, 1642033)-Net in Base 256 — Upper bound on s

There is no (25, 67, 1642034)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 224947 375139 525549 951029 091734 189702 758088 007401 365831 333477 542204 084866 797139 974660 358342 231747 715473 490345 263595 614961 714282 988612 498522 033700 685531 447560 312696 > 256⁶⁷ [i]