MinT - Best Known (59−21, 59, s)-Nets in Base 8

Best Known (59−21, 59, s)-Nets in Base 8

(59−21, 59, 354)-Net over F₈ — Constructive and digital

Digital (38, 59, 354)-net over F₈, using

3 times m-reduction [i] based on digital (38, 62, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 31, 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]

(59−21, 59, 514)-Net in Base 8 — Constructive

(38, 59, 514)-net in base 8, using

1 times m-reduction [i] based on (38, 60, 514)-net in base 8, using
- base change [i] based on digital (23, 45, 514)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (23, 46, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 23, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(59−21, 59, 570)-Net over F₈ — Digital

Digital (38, 59, 570)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁹, 570, F₈, 21) (dual of [570, 511, 22]-code), using
- 49 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 13 times 0, 1, 30 times 0) [i] based on linear OA(8⁵⁵, 517, F₈, 21) (dual of [517, 462, 22]-code), using
  - constructionÂ XX applied to C₁Â =Â C([510,18]), C₂Â =Â C([0,19]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,18]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([510,19]) [i] based on
    1. 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]
    2. linear OA(8⁵², 511, F₈, 20) (dual of [511, 459, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(8⁵⁵, 511, F₈, 21) (dual of [511, 456, 22]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,19}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    4. 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]
    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)

(59−21, 59, 111886)-Net in Base 8 — Upper bound on s

There is no (38, 59, 111887)-net in base 8, because

1 times m-reduction [i] would yield (38, 58, 111887)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23946 595103 113579 620076 189178 578048 690717 288180 357977 > 8⁵⁸ [i]

Best Known (59−21, 59, s)-Nets in Base 8

(59−21, 59, 354)-Net over F8 — Constructive and digital

(59−21, 59, 514)-Net in Base 8 — Constructive

(59−21, 59, 570)-Net over F8 — Digital

(59−21, 59, 111886)-Net in Base 8 — Upper bound on s

(59−21, 59, 354)-Net over F₈ — Constructive and digital

(59−21, 59, 570)-Net over F₈ — Digital