MinT - Best Known (42−10, 42, s)-Nets in Base 2

Best Known (42−10, 42, s)-Nets in Base 2

(42−10, 42, 77)-Net over F₂ — Constructive and digital

Digital (32, 42, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 12, 36)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (20, 30, 41)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 10, 20)-net over F₂, using
      - linear code embedding found in a computer search [i]
    2. digital (10, 20, 21)-net over F₂, using
      - linear code embedding found in a computer search [i]

(42−10, 42, 136)-Net over F₂ — Digital

Digital (32, 42, 136)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴², 136, F₂, 2, 10) (dual of [(136, 2), 230, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴², 272, F₂, 10) (dual of [272, 230, 11]-code), using
  - 1 times truncation [i] based on linear OA(2⁴³, 273, F₂, 11) (dual of [273, 230, 12]-code), using
    - constructionÂ XX applied to C₁Â =Â C([253,6]), C₂Â =Â C([0,8]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,6]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([253,8]) [i] based on
      1. 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]
      2. linear OA(2³³, 255, F₂, 9) (dual of [255, 222, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(2⁴¹, 255, F₂, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â {−2,−1,…,8}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      4. 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]
      5. 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]
      6. linear OA(2¹, 9, F₂, 1) (dual of [9, 8, 2]-code) (see above)

(42−10, 42, 873)-Net in Base 2 — Upper bound on s

There is no (32, 42, 874)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 4 422213 758546 > 2⁴² [i]

Best Known (42−10, 42, s)-Nets in Base 2

(42−10, 42, 77)-Net over F2 — Constructive and digital

(42−10, 42, 136)-Net over F2 — Digital

(42−10, 42, 873)-Net in Base 2 — Upper bound on s

(42−10, 42, 77)-Net over F₂ — Constructive and digital

(42−10, 42, 136)-Net over F₂ — Digital