MinT - Best Known (19, 19+28, s)-Nets in Base 256

Best Known (19, 19+28, s)-Nets in Base 256

(19, 19+28, 519)-Net over F₂₅₆ — Constructive and digital

Digital (19, 47, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 31, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19, 19+28, 743)-Net over F₂₅₆ — Digital

Digital (19, 47, 743)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴⁷, 743, F₂₅₆, 28) (dual of [743, 696, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(256⁴⁷, 771, F₂₅₆, 28) (dual of [771, 724, 29]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {245,246,…,272}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(19, 19+28, 2882187)-Net in Base 256 — Upper bound on s

There is no (19, 47, 2882188)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 153914 764126 329954 310272 584584 050077 698608 081896 020286 054509 714259 978399 126579 514280 522069 963935 819409 175830 411936 > 256⁴⁷ [i]