MinT - Best Known (105−20, 105, s)-Nets in Base 2

Best Known (105−20, 105, s)-Nets in Base 2

(105−20, 105, 260)-Net over F₂ — Constructive and digital

Digital (85, 105, 260)-net over F₂, using

2¹ times duplication [i] based on digital (84, 104, 260)-net over F₂, using
- trace code for nets [i] based on digital (6, 26, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(105−20, 105, 455)-Net over F₂ — Digital

Digital (85, 105, 455)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰⁵, 455, F₂, 2, 20) (dual of [(455, 2), 805, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁰⁵, 524, F₂, 2, 20) (dual of [(524, 2), 943, 21]-NRT-code), using
  - strength reduction [i] based on linear OOA(2¹⁰⁵, 524, F₂, 2, 21) (dual of [(524, 2), 943, 22]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2¹⁰⁵, 1048, F₂, 21) (dual of [1048, 943, 22]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(2¹⁰⁴, 1047, F₂, 21) (dual of [1047, 943, 22]-code), using
        adding a parity check bit [i] based on linear OA(2¹⁰³, 1046, F₂, 20) (dual of [1046, 943, 21]-code), using
        constructionÂ XX applied to C₁Â =Â C([1021,16]), C₂Â =Â C([1,18]), C₃Â =Â C₁Â +Â C₂Â =Â C([1,16]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1021,18]) [i] based on
        linear OA(2⁹¹, 1023, F₂, 19) (dual of [1023, 932, 20]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(2⁹⁰, 1023, F₂, 18) (dual of [1023, 933, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(2¹⁰¹, 1023, F₂, 21) (dual of [1023, 922, 22]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,18}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(2⁸⁰, 1023, F₂, 16) (dual of [1023, 943, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 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, 11, F₂, 1) (dual of [11, 10, 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)

(105−20, 105, 6543)-Net in Base 2 — Upper bound on s

There is no (85, 105, 6544)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 40 576324 357443 369879 960693 388779 > 2¹⁰⁵ [i]

Best Known (105−20, 105, s)-Nets in Base 2

(105−20, 105, 260)-Net over F2 — Constructive and digital

(105−20, 105, 455)-Net over F2 — Digital

(105−20, 105, 6543)-Net in Base 2 — Upper bound on s

(105−20, 105, 260)-Net over F₂ — Constructive and digital

(105−20, 105, 455)-Net over F₂ — Digital