MinT - Best Known (59−20, 59, s)-Nets in Base 8

Best Known (59−20, 59, s)-Nets in Base 8

(59−20, 59, 354)-Net over F₈ — Constructive and digital

Digital (39, 59, 354)-net over F₈, using

5 times m-reduction [i] based on digital (39, 64, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 32, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(59−20, 59, 516)-Net in Base 8 — Constructive

(39, 59, 516)-net in base 8, using

1 times m-reduction [i] based on (39, 60, 516)-net in base 8, using
- base change [i] based on digital (24, 45, 516)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (24, 46, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 23, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(59−20, 59, 732)-Net over F₈ — Digital

Digital (39, 59, 732)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁹, 732, F₈, 20) (dual of [732, 673, 21]-code), using
- 208 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 41 times 0, 1, 59 times 0, 1, 71 times 0) [i] based on linear OA(8⁵², 517, F₈, 20) (dual of [517, 465, 21]-code), using
  - constructionÂ XX applied to C₁Â =Â C([510,17]), C₂Â =Â C([0,18]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,17]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([510,18]) [i] based on
    1. linear OA(8⁴⁹, 511, F₈, 19) (dual of [511, 462, 20]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,17}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(8⁴⁹, 511, F₈, 19) (dual of [511, 462, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(8⁵², 511, F₈, 20) (dual of [511, 459, 21]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,18}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    4. linear OA(8⁴⁶, 511, F₈, 18) (dual of [511, 465, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    5. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    6. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code) (see above)

(59−20, 59, 137749)-Net in Base 8 — Upper bound on s

There is no (39, 59, 137750)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 191566 846565 356696 787983 540391 726734 538904 607338 365676 > 8⁵⁹ [i]

Best Known (59−20, 59, s)-Nets in Base 8

(59−20, 59, 354)-Net over F8 — Constructive and digital

(59−20, 59, 516)-Net in Base 8 — Constructive

(59−20, 59, 732)-Net over F8 — Digital

(59−20, 59, 137749)-Net in Base 8 — Upper bound on s

(59−20, 59, 354)-Net over F₈ — Constructive and digital

(59−20, 59, 732)-Net over F₈ — Digital