MinT - Best Known (58−7, 58, s)-Nets in Base 8

Best Known (58−7, 58, s)-Nets in Base 8

(58−7, 58, 5605921)-Net over F₈ — Constructive and digital

Digital (51, 58, 5605921)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁸, 5605921, F₈, 9, 7) (dual of [(5605921, 9), 50453231, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8⁵⁸, 5605922, F₈, 3, 7) (dual of [(5605922, 3), 16817708, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(8⁸, 13520, F₈, 3, 3) (dual of [(13520, 3), 40552, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(8⁸, 13520, F₈, 2, 3) (dual of [(13520, 2), 27032, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁸, 13520, F₈, 3) (dual of [13520, 13512, 4]-code or 13520-cap in PG(7,8)), using
        product of projective cap with cap in AG(4,Â 8) [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
    2. linear OOA(8⁵⁰, 5592402, F₈, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
      - trace code [i] based on linear OOA(64²⁵, 2796201, F₆₄, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(58−7, 58, large)-Net over F₈ — Digital

Digital (51, 58, large)-net over F₈, using

t-expansion [i] based on digital (49, 58, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁸, large, F₈, 9) (dual of [large, large−58, 10]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁵⁷, large, F₈, 9) (dual of [large, large−57, 10]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(58−7, 58, large)-Net in Base 8 — Upper bound on s

There is no (51, 58, large)-net in base 8, because

5 times m-reduction [i] would yield (51, 53, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]