MinT - Best Known (50−6, 50, s)-Nets in Base 8

Best Known (50−6, 50, s)-Nets in Base 8

(50−6, 50, 5605922)-Net over F₈ — Constructive and digital

Digital (44, 50, 5605922)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 8, 13520)-net over F₈, using
  - net defined by OOA [i] based on 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. digital (36, 42, 5592402)-net over F₈, using
  - trace code for nets [i] based on digital (15, 21, 2796201)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²¹, 2796201, F₆₄, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(50−6, 50, large)-Net over F₈ — Digital

Digital (44, 50, large)-net over F₈, using

8¹ times duplication [i] based on digital (43, 49, large)-net over F₈, using
- t-expansion [i] based on digital (42, 49, large)-net over F₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(50−6, 50, large)-Net in Base 8 — Upper bound on s

There is no (44, 50, large)-net in base 8, because

4 times m-reduction [i] would yield (44, 46, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]