MinT - Best Known (22−13, 22, s)-Nets in Base 256

Best Known (22−13, 22, s)-Nets in Base 256

(22−13, 22, 517)-Net over F₂₅₆ — Constructive and digital

Digital (9, 22, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 15, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(22−13, 22, 758)-Net over F₂₅₆ — Digital

Digital (9, 22, 758)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256²², 758, F₂₅₆, 13) (dual of [758, 736, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(256²², 771, F₂₅₆, 13) (dual of [771, 749, 14]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {121,122,…,133}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(22−13, 22, 3151529)-Net in Base 256 — Upper bound on s

There is no (9, 22, 3151530)-net in base 256, because

1 times m-reduction [i] would yield (9, 21, 3151530)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 374 144483 605002 991821 342925 458115 672930 123952 590276 > 256²¹ [i]