MinT - Best Known (20, 20+30, s)-Nets in Base 256

Best Known (20, 20+30, s)-Nets in Base 256

(20, 20+30, 519)-Net over F₂₅₆ — Constructive and digital

Digital (20, 50, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 33, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(20, 20+30, 714)-Net over F₂₅₆ — Digital

Digital (20, 50, 714)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁵⁰, 714, F₂₅₆, 30) (dual of [714, 664, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(256⁵⁰, 771, F₂₅₆, 30) (dual of [771, 721, 31]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {1,39,77,…,332}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(20, 20+30, 2683439)-Net in Base 256 — Upper bound on s

There is no (20, 50, 2683440)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 582251 684420 467797 701952 110983 396710 372286 313697 969916 464568 327559 820994 078546 752664 165748 544825 111125 698514 357970 804251 > 256⁵⁰ [i]