MinT - Best Known (17, 39, s)-Nets in Base 256

Best Known (17, 39, s)-Nets in Base 256

(17, 39, 520)-Net over F₂₅₆ — Constructive and digital

Digital (17, 39, 520)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 25, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆ (see above)

(17, 39, 1217)-Net over F₂₅₆ — Digital

Digital (17, 39, 1217)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256³⁹, 1217, F₂₅₆, 22) (dual of [1217, 1178, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(256³⁹, 1285, F₂₅₆, 22) (dual of [1285, 1246, 23]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,78,155,…,333}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(17, 39, 6649346)-Net in Base 256 — Upper bound on s

There is no (17, 39, 6649347)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 8343 703485 462201 249561 931123 756004 121208 480200 359397 597849 356931 189068 289076 833696 639876 814336 > 256³⁹ [i]