MinT - Best Known (10, 10+15, s)-Nets in Base 256

Best Known (10, 10+15, s)-Nets in Base 256

(10, 10+15, 517)-Net over F₂₅₆ — Constructive and digital

Digital (10, 25, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 17, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(10, 10+15, 615)-Net over F₂₅₆ — Digital

Digital (10, 25, 615)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256²⁵, 615, F₂₅₆, 15) (dual of [615, 590, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(256²⁵, 771, F₂₅₆, 15) (dual of [771, 746, 16]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {1,33,65,…,449}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(10, 10+15, 2394421)-Net in Base 256 — Upper bound on s

There is no (10, 25, 2394422)-net in base 256, because

1 times m-reduction [i] would yield (10, 24, 2394422)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 6277 105522 047008 781596 424007 944320 111096 381844 870002 851096 > 256²⁴ [i]