MinT - Best Known (12, 12+16, s)-Nets in Base 256

Best Known (12, 12+16, s)-Nets in Base 256

(12, 12+16, 518)-Net over F₂₅₆ — Constructive and digital

Digital (12, 28, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 18, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(12, 12+16, 1040)-Net over F₂₅₆ — Digital

Digital (12, 28, 1040)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256²⁸, 1040, F₂₅₆, 16) (dual of [1040, 1012, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(256²⁸, 1285, F₂₅₆, 16) (dual of [1285, 1257, 17]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,18,35,…,256}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(12, 12+16, 3962683)-Net in Base 256 — Upper bound on s

There is no (12, 28, 3962684)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 26 959978 849535 370862 030784 589633 334965 690662 236454 170813 366837 063111 > 256²⁸ [i]