MinT - Best Known (206−31, 206, s)-Nets in Base 2

Best Known (206−31, 206, s)-Nets in Base 2

(206−31, 206, 624)-Net over F₂ — Constructive and digital

Digital (175, 206, 624)-net over F₂, using

2² times duplication [i] based on digital (173, 204, 624)-net over F₂, using
- trace code for nets [i] based on digital (3, 34, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(206−31, 206, 2057)-Net over F₂ — Digital

Digital (175, 206, 2057)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁶, 2057, F₂, 4, 31) (dual of [(2057, 4), 8022, 32]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2²⁰², 2056, F₂, 4, 31) (dual of [(2056, 4), 8022, 32]-NRT-code), using
  - OOA 4-folding [i] based on linear OA(2²⁰², 8224, F₂, 31) (dual of [8224, 8022, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(26) [i] based on
      1. linear OA(2¹⁹⁶, 8192, F₂, 31) (dual of [8192, 7996, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(2¹⁷⁰, 8192, F₂, 27) (dual of [8192, 8022, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
      3. linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
        Reedâ€“Muller code RM(1,5) [i]
        caps in base bÂ =Â 2 [i]

(206−31, 206, 83508)-Net in Base 2 — Upper bound on s

There is no (175, 206, 83509)-net in base 2, because

1 times m-reduction [i] would yield (175, 205, 83509)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 51 429866 110081 514747 583257 735307 887971 130584 644779 870061 948880 > 2²⁰⁵ [i]