MinT - Best Known (55−13, 55, s)-Nets in Base 8

Best Known (55−13, 55, s)-Nets in Base 8

(55−13, 55, 1368)-Net over F₈ — Constructive and digital

Digital (42, 55, 1368)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁵, 1368, F₈, 13, 13) (dual of [(1368, 13), 17729, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8⁵⁵, 8209, F₈, 13) (dual of [8209, 8154, 14]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁵⁴, 8208, F₈, 13) (dual of [8208, 8154, 14]-code), using
    - trace code [i] based on linear OA(64²⁷, 4104, F₆₄, 13) (dual of [4104, 4077, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
        linear OA(64²⁵, 4096, F₆₄, 13) (dual of [4096, 4071, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(64¹⁹, 4096, F₆₄, 10) (dual of [4096, 4077, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(55−13, 55, 10415)-Net over F₈ — Digital

Digital (42, 55, 10415)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−13, 55, large)-Net in Base 8 — Upper bound on s

There is no (42, 55, large)-net in base 8, because

11 times m-reduction [i] would yield (42, 44, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]