MinT - Best Known (156−16, 156, s)-Nets in Base 8

Best Known (156−16, 156, s)-Nets in Base 8

(156−16, 156, 2099208)-Net over F₈ — Constructive and digital

Digital (140, 156, 2099208)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (26, 34, 2058)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 9)-net over F₈, using
      - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
        the rational function field F₈(x) [i]
        Niederreiter sequence [i]
    2. digital (22, 30, 2049)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8³⁰, 2049, F₈, 8, 8) (dual of [(2049, 8), 16362, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(8³⁰, 8196, F₈, 8) (dual of [8196, 8166, 9]-code), using
        trace code [i] based on linear OA(64¹⁵, 4098, F₆₄, 8) (dual of [4098, 4083, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(64¹⁵, 4096, F₆₄, 8) (dual of [4096, 4081, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹³, 4096, F₆₄, 7) (dual of [4096, 4083, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [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. digital (106, 122, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²², 2097150, F₈, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹²², 8388601, F₈, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹²², 8388602, F₈, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
        trace code [i] based on linear OOA(64⁶¹, 4194301, F₆₄, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶¹, 8388602, F₆₄, 16) (dual of [8388602, 8388541, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(156−16, 156, large)-Net over F₈ — Digital

Digital (140, 156, large)-net over F₈, using

t-expansion [i] based on digital (138, 156, large)-net over F₈, using
- 5 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(156−16, 156, large)-Net in Base 8 — Upper bound on s

There is no (140, 156, large)-net in base 8, because

14 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (156−16, 156, s)-Nets in Base 8

(156−16, 156, 2099208)-Net over F8 — Constructive and digital

(156−16, 156, large)-Net over F8 — Digital

(156−16, 156, large)-Net in Base 8 — Upper bound on s

(156−16, 156, 2099208)-Net over F₈ — Constructive and digital

(156−16, 156, large)-Net over F₈ — Digital