MinT - Best Known (86−22, 86, s)-Nets in Base 8

Best Known (86−22, 86, s)-Nets in Base 8

(86−22, 86, 745)-Net over F₈ — Constructive and digital

Digital (64, 86, 745)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁸⁶, 745, F₈, 22, 22) (dual of [(745, 22), 16304, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8⁸⁶, 8195, F₈, 22) (dual of [8195, 8109, 23]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁸⁶, 8196, F₈, 22) (dual of [8196, 8110, 23]-code), using
    - trace code [i] based on linear OA(64⁴³, 4098, F₆₄, 22) (dual of [4098, 4055, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
        linear OA(64⁴³, 4096, F₆₄, 22) (dual of [4096, 4053, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(64⁴¹, 4096, F₆₄, 21) (dual of [4096, 4055, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(86−22, 86, 772)-Net in Base 8 — Constructive

(64, 86, 772)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (15, 26, 258)-net in base 8, using
  - trace code for nets [i] based on (2, 13, 129)-net in base 64, using
    - 1 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
      - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. (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]

(86−22, 86, 8160)-Net over F₈ — Digital

Digital (64, 86, 8160)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁶, 8160, F₈, 22) (dual of [8160, 8074, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁶, 8192, F₈, 22) (dual of [8192, 8106, 23]-code), using
  - trace code [i] based on linear OA(64⁴³, 4096, F₆₄, 22) (dual of [4096, 4053, 23]-code), using
    - an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(86−22, 86, 8061901)-Net in Base 8 — Upper bound on s

There is no (64, 86, 8061902)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 463168 778579 959202 799364 875481 038986 097855 897069 347747 957545 194362 529359 877890 > 8⁸⁶ [i]