MinT - Best Known (123−7, 123, s)-Nets in Base 8

Best Known (123−7, 123, s)-Nets in Base 8

(123−7, 123, large)-Net over F₈ — Constructive and digital

Digital (116, 123, large)-net over F₈, using

4 times m-reduction [i] based on digital (116, 127, large)-net over F₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (28, 33, 3355440)-net over F₈, using
    - s-reduction based on digital (28, 33, 4194301)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8³³, 4194301, F₈, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(8³³, large, F₈, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
  3. 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]

(123−7, 123, large)-Net in Base 8 — Upper bound on s

There is no (116, 123, large)-net in base 8, because

5 times m-reduction [i] would yield (116, 118, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (123−7, 123, s)-Nets in Base 8

(123−7, 123, large)-Net over F8 — Constructive and digital

(123−7, 123, large)-Net in Base 8 — Upper bound on s

(123−7, 123, large)-Net over F₈ — Constructive and digital