MinT - Best Known (127−5, 127, s)-Nets in Base 8

Best Known (127−5, 127, s)-Nets in Base 8

(127−5, 127, large)-Net over F₈ — Constructive and digital

Digital (122, 127, large)-net over F₈, using

6 times m-reduction [i] based on digital (122, 133, large)-net over F₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 1048579)-net over F₈, using
    - s-reduction based on digital (0, 1, s)-net over F₈ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  2. digital (0, 1, 1048579)-net over F₈ (see above)
  3. digital (0, 1, 1048579)-net over F₈ (see above)
  4. digital (6, 8, 1048579)-net over F₈, using
    - s-reduction based on digital (6, 8, 2396745)-net over F₈, using
      - kÂ =Â 2 construction [i]
  5. digital (6, 8, 1048579)-net over F₈ (see above)
  6. digital (9, 12, 1048579)-net over F₈, using
    - s-reduction based on digital (9, 12, 2931457)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8¹², 2931457, F₈, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8¹², 2931457, F₈, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8¹², 2931457, F₈, 3) (dual of [2931457, 2931445, 4]-code or 2931457-cap in PG(11,8)), using
        product of projective cap with cap in AG(5,Â 8) [i] based on linear OA(8⁷, 4224, F₈, 3) (dual of [4224, 4217, 4]-code or 4224-cap in PG(6,8)), using
        product of two caps with tangent hyperplane [i]
  7. digital (24, 29, 1048579)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8²⁹, 1048579, F₈, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(8²⁹, 2097159, F₈, 5) (dual of [2097159, 2097130, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(8²⁹, 2097152, F₈, 5) (dual of [2097152, 2097123, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8²², 2097152, F₈, 4) (dual of [2097152, 2097130, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8⁰, 7, F₈, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  8. digital (62, 73, 1677720)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁷³, 1677720, F₈, 11, 11) (dual of [(1677720, 11), 18454847, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(8⁷³, 8388601, F₈, 11) (dual of [8388601, 8388528, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷³, large, F₈, 11) (dual of [large, large−73, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(127−5, 127, large)-Net in Base 8 — Upper bound on s

There is no (122, 127, large)-net in base 8, because

3 times m-reduction [i] would yield (122, 124, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (127−5, 127, s)-Nets in Base 8

(127−5, 127, large)-Net over F8 — Constructive and digital

(127−5, 127, large)-Net in Base 8 — Upper bound on s

(127−5, 127, large)-Net over F₈ — Constructive and digital