MinT - Best Known (31−4, 31, s)-Nets in Base 8

Best Known (31−4, 31, s)-Nets in Base 8

(31−4, 31, large)-Net over F₈ — Constructive and digital

Digital (27, 31, large)-net over F₈, using

8³ times duplication [i] based on digital (24, 28, large)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8²⁸, large, F₈, 4, 4), using
  - appending kth column [i] based on linear OOA(8²⁸, large, F₈, 3, 4), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(8², 9, F₈, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;25,8) [i]
      2. 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]

(31−4, 31, large)-Net in Base 8 — Upper bound on s

There is no (27, 31, large)-net in base 8, because

2 times m-reduction [i] would yield (27, 29, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]