MinT - Best Known (69, 69+23, s)-Nets in Base 8

Best Known (69, 69+23, s)-Nets in Base 8

(69, 69+23, 745)-Net over F₈ — Constructive and digital

Digital (69, 92, 745)-net over F₈, using

8² times duplication [i] based on digital (67, 90, 745)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁹⁰, 745, F₈, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(8⁹⁰, 8196, F₈, 23) (dual of [8196, 8106, 24]-code), using
    - trace code [i] based on linear OA(64⁴⁵, 4098, F₆₄, 23) (dual of [4098, 4053, 24]-code), using
      - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
        linear OA(64⁴⁵, 4096, F₆₄, 23) (dual of [4096, 4051, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        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(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]

(69, 69+23, 1028)-Net in Base 8 — Constructive

(69, 92, 1028)-net in base 8, using

base change [i] based on digital (46, 69, 1028)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (45, 68, 1028)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 22, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 11, 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 (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 (see above)

(69, 69+23, 8204)-Net over F₈ — Digital

Digital (69, 92, 8204)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹², 8204, F₈, 23) (dual of [8204, 8112, 24]-code), using
- trace code [i] based on linear OA(64⁴⁶, 4102, F₆₄, 23) (dual of [4102, 4056, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. 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]
    2. linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(69, 69+23, large)-Net in Base 8 — Upper bound on s

There is no (69, 92, large)-net in base 8, because

21 times m-reduction [i] would yield (69, 71, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]