MinT - Best Known (35−8, 35, s)-Nets in Base 2

Best Known (35−8, 35, s)-Nets in Base 2

(35−8, 35, 78)-Net over F₂ — Constructive and digital

Digital (27, 35, 78)-net over F₂, using

2³ 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]

(35−8, 35, 145)-Net over F₂ — Digital

Digital (27, 35, 145)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³⁵, 145, F₂, 8, 8) (dual of [(145, 8), 1125, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³⁵, 145, F₂, 7, 8) (dual of [(145, 7), 980, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2³⁵, 145, F₂, 8) (dual of [145, 110, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(2³⁵, 274, F₂, 8) (dual of [274, 239, 9]-code), using
      - constructionÂ XX applied to C₁Â =Â C([253,4]), C₂Â =Â C([1,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([1,4]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([253,6]) [i] based on
        linear OA(2²⁵, 255, F₂, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â {−2,−1,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(2²⁴, 255, F₂, 6) (dual of [255, 231, 7]-code), using 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(2³³, 255, F₂, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â {−2,−1,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(2¹⁶, 255, F₂, 4) (dual of [255, 239, 5]-code), using 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(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]
        linear OA(2¹, 9, F₂, 1) (dual of [9, 8, 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 (see above)

(35−8, 35, 947)-Net in Base 2 — Upper bound on s

There is no (27, 35, 948)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 34438 757324 > 2³⁵ [i]

Best Known (35−8, 35, s)-Nets in Base 2

(35−8, 35, 78)-Net over F2 — Constructive and digital

(35−8, 35, 145)-Net over F2 — Digital

(35−8, 35, 947)-Net in Base 2 — Upper bound on s

(35−8, 35, 78)-Net over F₂ — Constructive and digital

(35−8, 35, 145)-Net over F₂ — Digital