MinT - Best Known (120−13, 120, s)-Nets in Base 8

Best Known (120−13, 120, s)-Nets in Base 8

(120−13, 120, 2798932)-Net over F₈ — Constructive and digital

Digital (107, 120, 2798932)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (16, 22, 2732)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²², 2732, F₈, 6, 6) (dual of [(2732, 6), 16370, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(8²², 8196, F₈, 6) (dual of [8196, 8174, 7]-code), using
      - trace code [i] based on linear OA(64¹¹, 4098, F₆₄, 6) (dual of [4098, 4087, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        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(64⁹, 4096, F₆₄, 5) (dual of [4096, 4087, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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 (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(120−13, 120, large)-Net over F₈ — Digital

Digital (107, 120, large)-net over F₈, using

t-expansion [i] based on digital (106, 120, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (106, 124, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁴, large, F₈, 18) (dual of [large, large−124, 19]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²¹, large, F₈, 18) (dual of [large, large−121, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(120−13, 120, large)-Net in Base 8 — Upper bound on s

There is no (107, 120, large)-net in base 8, because

11 times m-reduction [i] would yield (107, 109, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (120−13, 120, s)-Nets in Base 8

(120−13, 120, 2798932)-Net over F8 — Constructive and digital

(120−13, 120, large)-Net over F8 — Digital

(120−13, 120, large)-Net in Base 8 — Upper bound on s

(120−13, 120, 2798932)-Net over F₈ — Constructive and digital

(120−13, 120, large)-Net over F₈ — Digital