MinT - Best Known (17−4, 17, s)-Nets in Base 2

Best Known (17−4, 17, s)-Nets in Base 2

(17−4, 17, 257)-Net over F₂ — Constructive and digital

Digital (13, 17, 257)-net over F₂, using

nets constructed from net-embeddable BCH codes [i]

(17−4, 17, 265)-Net over F₂ — Digital

Digital (13, 17, 265)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁷, 265, F₂, 4, 4) (dual of [(265, 4), 1043, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁷, 265, F₂, 3, 4) (dual of [(265, 3), 778, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2¹⁷, 265, F₂, 4) (dual of [265, 248, 5]-code), using
    - 1 times truncation [i] based on linear OA(2¹⁸, 266, F₂, 5) (dual of [266, 248, 6]-code), using
      - constructionÂ X4 applied to C_e(4) âŠ‚ C_e(2) [i] based on
        linear OA(2¹⁷, 256, F₂, 5) (dual of [256, 239, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(2⁹, 256, F₂, 3) (dual of [256, 247, 4]-code or 256-cap in PG(8,2)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(2⁹, 10, F₂, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,2)), using
        dual of repetition code with length 10 [i]
        linear OA(2¹, 10, F₂, 1) (dual of [10, 9, 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]

(17−4, 17, 509)-Net in Base 2 — Upper bound on s

There is no (13, 17, 510)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 131326 > 2¹⁷ [i]