MinT - Best Known (56−34, 56, s)-Nets in Base 256

Best Known (56−34, 56, s)-Nets in Base 256

(56−34, 56, 519)-Net over F₂₅₆ — Constructive and digital

Digital (22, 56, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 37, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(56−34, 56, 677)-Net over F₂₅₆ — Digital

Digital (22, 56, 677)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁵⁶, 677, F₂₅₆, 34) (dual of [677, 621, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(256⁵⁶, 771, F₂₅₆, 34) (dual of [771, 715, 35]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {1,219,437,…,256}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(56−34, 56, 2412289)-Net in Base 256 — Upper bound on s

There is no (22, 56, 2412290)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 726 843684 633911 532309 971403 057913 817529 899957 013333 766555 741627 910726 295409 916979 833874 293685 466071 309496 126554 489374 158066 376252 705776 > 256⁵⁶ [i]