MinT - Best Known (100−25, 100, s)-Nets in Base 8

Best Known (100−25, 100, s)-Nets in Base 8

(100−25, 100, 683)-Net over F₈ — Constructive and digital

Digital (75, 100, 683)-net over F₈, using

8¹ times duplication [i] based on digital (74, 99, 683)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁹⁹, 683, F₈, 25, 25) (dual of [(683, 25), 16976, 26]-NRT-code), using
  - OOA 12-folding and stacking with additional row [i] based on linear OA(8⁹⁹, 8197, F₈, 25) (dual of [8197, 8098, 26]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(8⁹⁸, 8196, F₈, 25) (dual of [8196, 8098, 26]-code), using
      - trace code [i] based on linear OA(64⁴⁹, 4098, F₆₄, 25) (dual of [4098, 4049, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(23) [i] based on
        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(64⁴⁷, 4096, F₆₄, 24) (dual of [4096, 4049, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [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]

(100−25, 100, 1028)-Net in Base 8 — Constructive

(75, 100, 1028)-net in base 8, using

base change [i] based on digital (50, 75, 1028)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (49, 74, 1028)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (12, 24, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 12, 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]
    2. digital (25, 50, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(100−25, 100, 8204)-Net over F₈ — Digital

Digital (75, 100, 8204)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁰, 8204, F₈, 25) (dual of [8204, 8104, 26]-code), using
- trace code [i] based on linear OA(64⁵⁰, 4102, F₆₄, 25) (dual of [4102, 4052, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,11]) [i] based on
    1. linear OA(64⁴⁹, 4097, F₆₄, 25) (dual of [4097, 4048, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(64⁴⁵, 4097, F₆₄, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    3. linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(100−25, 100, large)-Net in Base 8 — Upper bound on s

There is no (75, 100, large)-net in base 8, because

23 times m-reduction [i] would yield (75, 77, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (100−25, 100, s)-Nets in Base 8

(100−25, 100, 683)-Net over F8 — Constructive and digital

(100−25, 100, 1028)-Net in Base 8 — Constructive

(100−25, 100, 8204)-Net over F8 — Digital

(100−25, 100, large)-Net in Base 8 — Upper bound on s

(100−25, 100, 683)-Net over F₈ — Constructive and digital

(100−25, 100, 8204)-Net over F₈ — Digital