MinT - Best Known (55−33, 55, s)-Nets in Base 256

Best Known (55−33, 55, s)-Nets in Base 256

(55−33, 55, 520)-Net over F₂₅₆ — Constructive and digital

Digital (22, 55, 520)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 36, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆ (see above)

(55−33, 55, 749)-Net over F₂₅₆ — Digital

Digital (22, 55, 749)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁵⁵, 749, F₂₅₆, 33) (dual of [749, 694, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(256⁵⁵, 771, F₂₅₆, 33) (dual of [771, 716, 34]-code), using
  - the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {1,17,33,…,513}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(55−33, 55, 3579377)-Net in Base 256 — Upper bound on s

There is no (22, 55, 3579378)-net in base 256, because

1 times m-reduction [i] would yield (22, 54, 3579378)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 11090 701230 702356 181388 957571 593150 034011 794845 986075 577910 583580 814050 773226 876548 085683 030519 526783 567668 162648 181647 170679 061491 > 256⁵⁴ [i]