MinT - Best Known (239−15, 239, s)-Nets in Base 2

Best Known (239−15, 239, s)-Nets in Base 2

(239−15, 239, 2396768)-Net over F₂ — Constructive and digital

Digital (224, 239, 2396768)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (58, 65, 2097150)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2⁶⁵, 2097150, F₂, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
      - OOAs constructed from BCH codes [i]
2. digital (159, 174, 1198384)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 12, 13)-net over F₂, using
      - linear code embedding found in a computer search [i]
    2. digital (147, 162, 1198371)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2¹⁶², 1198371, F₂, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
        OOA 7-folding and stacking with additional row [i] based on linear OA(2¹⁶², 8388598, F₂, 15) (dual of [8388598, 8388436, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(239−15, 239, 5592414)-Net over F₂ — Digital

Digital (224, 239, 5592414)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁹, 5592414, F₂, 3, 15) (dual of [(5592414, 3), 16777003, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁶⁸, 4194303, F₂, 3, 7) (dual of [(4194303, 3), 12582841, 8]-NRT-code), using
    - OOAs constructed from BCH codes [i]
  2. linear OOA(2¹⁷¹, 2796207, F₂, 3, 15) (dual of [(2796207, 3), 8388450, 16]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁹, 6, F₂, 3, 7) (dual of [(6, 3), 9, 8]-NRT-code), using
        extended algebraic-geometric NRT-code AG_e(3;F,10P) [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
        an explicitly constructive algebraic function field [i]
      2. linear OOA(2¹⁶², 2796201, F₂, 3, 15) (dual of [(2796201, 3), 8388441, 16]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(239−15, 239, large)-Net in Base 2 — Upper bound on s

There is no (224, 239, large)-net in base 2, because

13 times m-reduction [i] would yield (224, 226, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]