MinT - Best Known (34−8, 34, s)-Nets in Base 2

Best Known (34−8, 34, s)-Nets in Base 2

(34−8, 34, 78)-Net over F₂ — Constructive and digital

Digital (26, 34, 78)-net over F₂, using

2² 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]

(34−8, 34, 136)-Net over F₂ — Digital

Digital (26, 34, 136)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³⁴, 136, F₂, 8, 8) (dual of [(136, 8), 1054, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³⁴, 136, F₂, 7, 8) (dual of [(136, 7), 918, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³⁴, 136, F₂, 2, 8) (dual of [(136, 2), 238, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2³⁴, 272, F₂, 8) (dual of [272, 238, 9]-code), using
      - 1 times truncation [i] based on linear OA(2³⁵, 273, F₂, 9) (dual of [273, 238, 10]-code), using
        constructionÂ XX applied to C₁Â =Â C([253,4]), C₂Â =Â C([0,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,4]), 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(2²⁵, 255, F₂, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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(2¹⁷, 255, F₂, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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, using
        parity-check code [i]
        linear OA(2¹, 9, F₂, 1) (dual of [9, 8, 2]-code) (see above)

(34−8, 34, 795)-Net in Base 2 — Upper bound on s

There is no (26, 34, 796)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 17193 678606 > 2³⁴ [i]

Best Known (34−8, 34, s)-Nets in Base 2

(34−8, 34, 78)-Net over F2 — Constructive and digital

(34−8, 34, 136)-Net over F2 — Digital

(34−8, 34, 795)-Net in Base 2 — Upper bound on s

(34−8, 34, 78)-Net over F₂ — Constructive and digital

(34−8, 34, 136)-Net over F₂ — Digital