MinT - Best Known (27−4, 27, s)-Nets in Base 8

Best Known (27−4, 27, s)-Nets in Base 8

(27−4, 27, 8388602)-Net over F₈ — Constructive and digital

Digital (23, 27, 8388602)-net over F₈, using

8¹ times duplication [i] based on digital (22, 26, 8388602)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8²⁶, 8388602, F₈, 4, 4) (dual of [(8388602, 4), 33554382, 5]-NRT-code), using
  - appending kth column [i] based on linear OOA(8²⁶, 8388602, F₈, 3, 4) (dual of [(8388602, 3), 25165780, 5]-NRT-code), using
    - trace code [i] based on linear OOA(64¹³, 4194301, F₆₄, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(64¹³, 8388602, F₆₄, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹³, large, F₆₄, 4) (dual of [large, large−13, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(27−4, 27, large)-Net over F₈ — Digital

Digital (23, 27, large)-net over F₈, using

8² times duplication [i] based on digital (21, 25, large)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8²⁵, large, F₈, 4, 4), using
  - appending kth column [i] based on linear OOA(8²⁵, large, F₈, 3, 4), using
    - Gilbertâ€“VarÅ¡amov bound [i]

(27−4, 27, large)-Net in Base 8 — Upper bound on s

There is no (23, 27, large)-net in base 8, because

2 times m-reduction [i] would yield (23, 25, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]