MinT - Best Known (82−11, 82, s)-Nets in Base 27

Best Known (82−11, 82, s)-Nets in Base 27

(82−11, 82, 5033160)-Net over F₂₇ — Constructive and digital

Digital (71, 82, 5033160)-net over F₂₇, using

27¹ times duplication [i] based on digital (70, 81, 5033160)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 9, 1677720)-net over F₂₇, using
    - s-reduction based on digital (6, 9, large)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁹, large, F₂₇, 3, 3), using
        appending kth column [i] based on linear OOA(27⁹, large, F₂₇, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁹, large, F₂₇, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
  2. digital (16, 21, 1677720)-net over F₂₇, using
    - s-reduction based on digital (16, 21, 4194301)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27²¹, 4194301, F₂₇, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27²¹, large, F₂₇, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’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 (40, 51, 1677720)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(82−11, 82, large)-Net over F₂₇ — Digital

Digital (71, 82, large)-net over F₂₇, using

t-expansion [i] based on digital (68, 82, large)-net over F₂₇, using
- 4 times m-reduction [i] based on digital (68, 86, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸⁶, large, F₂₇, 18) (dual of [large, large−86, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(82−11, 82, large)-Net in Base 27 — Upper bound on s

There is no (71, 82, large)-net in base 27, because

9 times m-reduction [i] would yield (71, 73, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (82−11, 82, s)-Nets in Base 27

(82−11, 82, 5033160)-Net over F27 — Constructive and digital

(82−11, 82, large)-Net over F27 — Digital

(82−11, 82, large)-Net in Base 27 — Upper bound on s

(82−11, 82, 5033160)-Net over F₂₇ — Constructive and digital

(82−11, 82, large)-Net over F₂₇ — Digital