MinT - Best Known (103−26, 103, s)-Nets in Base 8

Best Known (103−26, 103, s)-Nets in Base 8

(103−26, 103, 630)-Net over F₈ — Constructive and digital

Digital (77, 103, 630)-net over F₈, using

8¹ times duplication [i] based on digital (76, 102, 630)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁰², 630, F₈, 26, 26) (dual of [(630, 26), 16278, 27]-NRT-code), using
  - OA 13-folding and stacking [i] based on linear OA(8¹⁰², 8190, F₈, 26) (dual of [8190, 8088, 27]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹⁰², 8192, F₈, 26) (dual of [8192, 8090, 27]-code), using
      - trace code [i] based on linear OA(64⁵¹, 4096, F₆₄, 26) (dual of [4096, 4045, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(103−26, 103, 772)-Net in Base 8 — Constructive

(77, 103, 772)-net in base 8, using

1 times m-reduction [i] based on (77, 104, 772)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (19, 32, 258)-net in base 8, using
    - trace code for nets [i] based on (3, 16, 129)-net in base 64, using
      - 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
        base change [i] based on digital (0, 18, 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. (45, 72, 514)-net in base 8, using
    - base change [i] based on digital (27, 54, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 27, 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]

(103−26, 103, 8198)-Net over F₈ — Digital

Digital (77, 103, 8198)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰³, 8198, F₈, 26) (dual of [8198, 8095, 27]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8¹⁰², 8196, F₈, 26) (dual of [8196, 8094, 27]-code), using
    - trace code [i] based on linear OA(64⁵¹, 4098, F₆₄, 26) (dual of [4098, 4047, 27]-code), using
      - constructionÂ X applied to C_e(25) âŠ‚ C_e(24) [i] based on
        linear OA(64⁵¹, 4096, F₆₄, 26) (dual of [4096, 4045, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
        linear OA(64⁴⁹, 4096, F₆₄, 25) (dual of [4096, 4047, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]
  2. linear OA(8¹⁰², 8197, F₈, 25) (dual of [8197, 8095, 26]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹⁰²Â > V_b^sâˆ’1(kâˆ’1)Â = 2 521375 082338 364456 404226 842126 278738 853375 056216 413534 096799 626528 764135 982447 256885 859328 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 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]

(103−26, 103, large)-Net in Base 8 — Upper bound on s

There is no (77, 103, large)-net in base 8, because

24 times m-reduction [i] would yield (77, 79, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (103−26, 103, s)-Nets in Base 8

(103−26, 103, 630)-Net over F8 — Constructive and digital

(103−26, 103, 772)-Net in Base 8 — Constructive

(103−26, 103, 8198)-Net over F8 — Digital

(103−26, 103, large)-Net in Base 8 — Upper bound on s

(103−26, 103, 630)-Net over F₈ — Constructive and digital

(103−26, 103, 8198)-Net over F₈ — Digital