MinT - Best Known (15, 15+22, s)-Nets in Base 256

Best Known (15, 15+22, s)-Nets in Base 256

(15, 15+22, 518)-Net over F₂₅₆ — Constructive and digital

Digital (15, 37, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 24, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(15, 15+22, 695)-Net over F₂₅₆ — Digital

Digital (15, 37, 695)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256³⁷, 695, F₂₅₆, 22) (dual of [695, 658, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(256³⁷, 771, F₂₅₆, 22) (dual of [771, 734, 23]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {248,249,…,269}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(15, 15+22, 2426144)-Net in Base 256 — Upper bound on s

There is no (15, 37, 2426145)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 127315 266046 605794 863751 671955 189637 337365 617163 457764 521540 550881 654758 732784 543322 162976 > 256³⁷ [i]