MinT - Best Known (22, 22+32, s)-Nets in Base 256

Best Known (22, 22+32, s)-Nets in Base 256

(22, 22+32, 520)-Net over F₂₅₆ — Constructive and digital

Digital (22, 54, 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, 35, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆ (see above)

(22, 22+32, 793)-Net over F₂₅₆ — Digital

Digital (22, 54, 793)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁵⁴, 793, F₂₅₆, 32) (dual of [793, 739, 33]-code), using
- 19 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 18 times 0) [i] based on linear OA(256⁵³, 773, F₂₅₆, 32) (dual of [773, 720, 33]-code), using
  - constructionÂ X applied to C([241,272]) âŠ‚ C([242,272]) [i] based on
    1. linear OA(256⁵³, 771, F₂₅₆, 32) (dual of [771, 718, 33]-code), using the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {241,242,…,272}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(256⁵¹, 771, F₂₅₆, 31) (dual of [771, 720, 32]-code), using the BCH-code C(I) with length 771Â | 256²âˆ’1, defining interval IÂ =Â {242,243,…,272}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(22, 22+32, 3579377)-Net in Base 256 — Upper bound on s

There is no (22, 54, 3579378)-net in base 256, because

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]