MinT - Best Known (82−22, 82, s)-Nets in Base 4

Best Known (82−22, 82, s)-Nets in Base 4

(82−22, 82, 384)-Net over F₄ — Constructive and digital

Digital (60, 82, 384)-net over F₄, using

4¹ times duplication [i] based on digital (59, 81, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 27, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(82−22, 82, 450)-Net in Base 4 — Constructive

(60, 82, 450)-net in base 4, using

4¹ times duplication [i] based on (59, 81, 450)-net in base 4, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
  - 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
    - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(82−22, 82, 745)-Net over F₄ — Digital

Digital (60, 82, 745)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸², 745, F₄, 22) (dual of [745, 663, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸², 1034, F₄, 22) (dual of [1034, 952, 23]-code), using
  - constructionÂ XX applied to C₁Â =Â C([321,341]), C₂Â =Â C([323,342]), C₃Â =Â C₁Â +Â C₂Â =Â C([323,341]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([321,342]) [i] based on
    1. linear OA(4⁷⁶, 1023, F₄, 21) (dual of [1023, 947, 22]-code), using the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {321,322,…,341}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(4⁷⁶, 1023, F₄, 20) (dual of [1023, 947, 21]-code), using the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {323,324,…,342}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(4⁸¹, 1023, F₄, 22) (dual of [1023, 942, 23]-code), using the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {321,322,…,342}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    4. linear OA(4⁷¹, 1023, F₄, 19) (dual of [1023, 952, 20]-code), using the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {323,324,…,341}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    5. linear OA(4¹, 6, F₄, 1) (dual of [6, 5, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    6. linear OA(4⁰, 5, F₄, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(82−22, 82, 50338)-Net in Base 4 — Upper bound on s

There is no (60, 82, 50339)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23 385726 107207 625176 316471 437235 609262 876422 917104 > 4⁸² [i]

Best Known (82−22, 82, s)-Nets in Base 4

(82−22, 82, 384)-Net over F4 — Constructive and digital

(82−22, 82, 450)-Net in Base 4 — Constructive

(82−22, 82, 745)-Net over F4 — Digital

(82−22, 82, 50338)-Net in Base 4 — Upper bound on s

(82−22, 82, 384)-Net over F₄ — Constructive and digital

(82−22, 82, 745)-Net over F₄ — Digital