MinT - Best Known (38−22, 38, s)-Nets in Base 256

Best Known (38−22, 38, s)-Nets in Base 256

(38−22, 38, 519)-Net over F₂₅₆ — Constructive and digital

Digital (16, 38, 519)-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 (3, 25, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−22, 38, 819)-Net over F₂₅₆ — Digital

Digital (16, 38, 819)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256³⁸, 819, F₂₅₆, 22) (dual of [819, 781, 23]-code), using
- 46 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 45 times 0) [i] based on linear OA(256³⁶, 771, F₂₅₆, 22) (dual of [771, 735, 23]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {256,354,452,…,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(38−22, 38, 4016500)-Net in Base 256 — Upper bound on s

There is no (16, 38, 4016501)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 32 592594 901257 483647 052938 354691 436987 056265 355474 320767 903490 288589 305166 600540 245325 876056 > 256³⁸ [i]