MinT - Best Known (108−11, 108, s)-Nets in Base 8

Best Known (108−11, 108, s)-Nets in Base 8

(108−11, 108, 3617586)-Net over F₈ — Constructive and digital

Digital (97, 108, 3617586)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 26, 262146)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁶, 262146, F₈, 5, 5) (dual of [(262146, 5), 1310704, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(8²⁶, 524293, F₈, 5) (dual of [524293, 524267, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²⁶, 524294, F₈, 5) (dual of [524294, 524268, 6]-code), using
        trace code [i] based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 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 (71, 82, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(108−11, 108, large)-Net over F₈ — Digital

Digital (97, 108, large)-net over F₈, using

t-expansion [i] based on digital (96, 108, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (96, 112, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², large, F₈, 16) (dual of [large, large−112, 17]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(108−11, 108, large)-Net in Base 8 — Upper bound on s

There is no (97, 108, large)-net in base 8, because

9 times m-reduction [i] would yield (97, 99, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (108−11, 108, s)-Nets in Base 8

(108−11, 108, 3617586)-Net over F8 — Constructive and digital

(108−11, 108, large)-Net over F8 — Digital

(108−11, 108, large)-Net in Base 8 — Upper bound on s

(108−11, 108, 3617586)-Net over F₈ — Constructive and digital

(108−11, 108, large)-Net over F₈ — Digital