MinT - Best Known (81, 81+12, s)-Nets in Base 8

Best Known (81, 81+12, s)-Nets in Base 8

(81, 81+12, 2796201)-Net over F₈ — Constructive and digital

Digital (81, 93, 2796201)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁹³, 2796201, F₈, 15, 12) (dual of [(2796201, 15), 41942922, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁹³, 5592403, F₈, 3, 12) (dual of [(5592403, 3), 16777116, 13]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(8⁹⁰, 5592402, F₈, 3, 12) (dual of [(5592402, 3), 16777116, 13]-NRT-code), using
    - trace code [i] based on linear OOA(64⁴⁵, 2796201, F₆₄, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(81, 81+12, large)-Net over F₈ — Digital

Digital (81, 93, large)-net over F₈, using

8⁴ times duplication [i] based on digital (77, 89, large)-net over F₈, using
- t-expansion [i] based on digital (76, 89, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁹, large, F₈, 13) (dual of [large, large−89, 14]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(81, 81+12, large)-Net in Base 8 — Upper bound on s

There is no (81, 93, large)-net in base 8, because

10 times m-reduction [i] would yield (81, 83, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]