MinT - Best Known (162−17, 162, s)-Nets in Base 8

Best Known (162−17, 162, s)-Nets in Base 8

(162−17, 162, 2099200)-Net over F₈ — Constructive and digital

Digital (145, 162, 2099200)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (24, 32, 2050)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8³², 2050, F₈, 8, 8) (dual of [(2050, 8), 16368, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(8³², 8200, F₈, 8) (dual of [8200, 8168, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(8³², 8202, F₈, 8) (dual of [8202, 8170, 9]-code), using
        trace code [i] based on linear OA(64¹⁶, 4101, F₆₄, 8) (dual of [4101, 4085, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [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(64¹¹, 4096, F₆₄, 6) (dual of [4096, 4085, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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]
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]

(162−17, 162, large)-Net over F₈ — Digital

Digital (145, 162, large)-net over F₈, using

t-expansion [i] based on digital (144, 162, large)-net over F₈, using
- 6 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(162−17, 162, large)-Net in Base 8 — Upper bound on s

There is no (145, 162, large)-net in base 8, because

15 times m-reduction [i] would yield (145, 147, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (162−17, 162, s)-Nets in Base 8

(162−17, 162, 2099200)-Net over F8 — Constructive and digital

(162−17, 162, large)-Net over F8 — Digital

(162−17, 162, large)-Net in Base 8 — Upper bound on s

(162−17, 162, 2099200)-Net over F₈ — Constructive and digital

(162−17, 162, large)-Net over F₈ — Digital