MinT - Best Known (160−17, 160, s)-Nets in Base 8

Best Known (160−17, 160, s)-Nets in Base 8

(160−17, 160, 2099199)-Net over F₈ — Constructive and digital

Digital (143, 160, 2099199)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. 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 (113, 130, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(160−17, 160, large)-Net over F₈ — Digital

Digital (143, 160, large)-net over F₈, using

t-expansion [i] based on digital (138, 160, large)-net over F₈, using
- 1 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]

(160−17, 160, large)-Net in Base 8 — Upper bound on s

There is no (143, 160, large)-net in base 8, because

15 times m-reduction [i] would yield (143, 145, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (160−17, 160, s)-Nets in Base 8

(160−17, 160, 2099199)-Net over F8 — Constructive and digital

(160−17, 160, large)-Net over F8 — Digital

(160−17, 160, large)-Net in Base 8 — Upper bound on s

(160−17, 160, 2099199)-Net over F₈ — Constructive and digital

(160−17, 160, large)-Net over F₈ — Digital