MinT - Best Known (40, 58, s)-Nets in Base 256

Best Known (40, 58, s)-Nets in Base 256

(40, 58, 932067)-Net over F₂₅₆ — Constructive and digital

Digital (40, 58, 932067)-net over F₂₅₆, using

t-expansion [i] based on digital (39, 58, 932067)-net over F₂₅₆, using
- net defined by OOA [i] based on linear OOA(256⁵⁸, 932067, F₂₅₆, 21, 19) (dual of [(932067, 21), 19573349, 20]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(256⁵⁸, 2796202, F₂₅₆, 3, 19) (dual of [(2796202, 3), 8388548, 20]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(256⁵⁵, 2796201, F₂₅₆, 3, 19) (dual of [(2796201, 3), 8388548, 20]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(256⁵⁵, large, F₂₅₆, 19) (dual of [large, large−55, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(40, 58, large)-Net over F₂₅₆ — Digital

Digital (40, 58, large)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁵⁸, large, F₂₅₆, 18) (dual of [large, large−58, 19]-code), using
- strength reduction [i] based on linear OA(256⁵⁸, large, F₂₅₆, 20) (dual of [large, large−58, 21]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(40, 58, large)-Net in Base 256 — Upper bound on s

There is no (40, 58, large)-net in base 256, because

16 times m-reduction [i] would yield (40, 42, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]