MinT - Best Known (131−23, 131, s)-Nets in Base 8

Best Known (131−23, 131, s)-Nets in Base 8

(131−23, 131, 23835)-Net over F₈ — Constructive and digital

Digital (108, 131, 23835)-net over F₈, using

8³ times duplication [i] based on digital (105, 128, 23835)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹²⁸, 23835, F₈, 23, 23) (dual of [(23835, 23), 548077, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(8¹²⁸, 262186, F₈, 23) (dual of [262186, 262058, 24]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹²⁸, 262188, F₈, 23) (dual of [262188, 262060, 24]-code), using
      - constructionÂ X applied to C([0,11]) âŠ‚ C([0,8]) [i] based on
        linear OA(8¹²¹, 262145, F₈, 23) (dual of [262145, 262024, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 8¹²âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(8⁸⁵, 262145, F₈, 17) (dual of [262145, 262060, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 8¹²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(8⁷, 43, F₈, 5) (dual of [43, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]

(131−23, 131, 308504)-Net over F₈ — Digital

Digital (108, 131, 308504)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−23, 131, large)-Net in Base 8 — Upper bound on s

There is no (108, 131, large)-net in base 8, because

21 times m-reduction [i] would yield (108, 110, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]