MinT - Best Known (46−26, 46, s)-Nets in Base 256

Best Known (46−26, 46, s)-Nets in Base 256

(46−26, 46, 521)-Net over F₂₅₆ — Constructive and digital

Digital (20, 46, 521)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 30, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(46−26, 46, 1249)-Net over F₂₅₆ — Digital

Digital (20, 46, 1249)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴⁶, 1249, F₂₅₆, 26) (dual of [1249, 1203, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(256⁴⁶, 1285, F₂₅₆, 26) (dual of [1285, 1239, 27]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,114,227,…,256}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(46−26, 46, 7384653)-Net in Base 256 — Upper bound on s

There is no (20, 46, 7384654)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 601 227210 622159 544143 960052 106813 895983 796352 116038 476278 689028 744428 451958 254258 056943 025423 993447 417724 447511 > 256⁴⁶ [i]