MinT - Best Known (25, 33, s)-Nets in Base 2

Best Known (25, 33, s)-Nets in Base 2

(25, 33, 78)-Net over F₂ — Constructive and digital

Digital (25, 33, 78)-net over F₂, using

2¹ times duplication [i] based on digital (24, 32, 78)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2³², 78, F₂, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
  - appending kth column [i] based on linear OOA(2³², 78, F₂, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2¹¹, 47, F₂, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
        appending 3 arbitrary columns [i] based on linear OOA(2¹¹, 47, F₂, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹¹, 47, F₂, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 11, 47)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. linear OOA(2²¹, 39, F₂, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (13, 21, 39)-net over F₂, using
        linear code embedding found in a computer search [i]

(25, 33, 132)-Net over F₂ — Digital

Digital (25, 33, 132)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³³, 132, F₂, 8, 8) (dual of [(132, 8), 1023, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³³, 132, F₂, 7, 8) (dual of [(132, 7), 891, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³³, 132, F₂, 2, 8) (dual of [(132, 2), 231, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2³³, 264, F₂, 8) (dual of [264, 231, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(2³³, 265, F₂, 8) (dual of [265, 232, 9]-code), using
        1 times truncation [i] based on linear OA(2³⁴, 266, F₂, 9) (dual of [266, 232, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2³³, 256, F₂, 9) (dual of [256, 223, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2²⁵, 256, F₂, 7) (dual of [256, 231, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [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]

(25, 33, 668)-Net in Base 2 — Upper bound on s

There is no (25, 33, 669)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 8623 200827 > 2³³ [i]

Best Known (25, 33, s)-Nets in Base 2

(25, 33, 78)-Net over F2 — Constructive and digital

(25, 33, 132)-Net over F2 — Digital

(25, 33, 668)-Net in Base 2 — Upper bound on s

(25, 33, 78)-Net over F₂ — Constructive and digital

(25, 33, 132)-Net over F₂ — Digital