MinT - Best Known (70−9, 70, s)-Nets in Base 2

Best Known (70−9, 70, s)-Nets in Base 2

(70−9, 70, 32772)-Net over F₂ — Constructive and digital

Digital (61, 70, 32772)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁷⁰, 32772, F₂, 9, 9) (dual of [(32772, 9), 294878, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁷⁰, 32772, F₂, 8, 9) (dual of [(32772, 8), 262106, 10]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OA(2⁷⁰, 131089, F₂, 9) (dual of [131089, 131019, 10]-code), using
    - discarding factors / shortening the dual code based on linear OA(2⁷⁰, 131090, F₂, 9) (dual of [131090, 131020, 10]-code), using
      - constructionÂ X applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁶⁹, 131072, F₂, 9) (dual of [131072, 131003, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 131071Â = 2¹⁷âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2⁵², 131072, F₂, 7) (dual of [131072, 131020, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 131071Â = 2¹⁷âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹, 18, F₂, 1) (dual of [18, 17, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(70−9, 70, 344995)-Net in Base 2 — Upper bound on s

There is no (61, 70, 344996)-net in base 2, because

1 times m-reduction [i] would yield (61, 69, 344996)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 590 299875 097752 521256 > 2⁶⁹ [i]