MinT - Best Known (45, 45+14, s)-Nets in Base 8

Best Known (45, 45+14, s)-Nets in Base 8

(45, 45+14, 1172)-Net over F₈ — Constructive and digital

Digital (45, 59, 1172)-net over F₈, using

8¹ times duplication [i] based on digital (44, 58, 1172)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁵⁸, 1172, F₈, 14, 14) (dual of [(1172, 14), 16350, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(8⁵⁸, 8204, F₈, 14) (dual of [8204, 8146, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁵⁸, 8208, F₈, 14) (dual of [8208, 8150, 15]-code), using
      - trace code [i] based on linear OA(64²⁹, 4104, F₆₄, 14) (dual of [4104, 4075, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(10) [i] based on
        linear OA(64²⁷, 4096, F₆₄, 14) (dual of [4096, 4069, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(64²¹, 4096, F₆₄, 11) (dual of [4096, 4075, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(45, 45+14, 10168)-Net over F₈ — Digital

Digital (45, 59, 10168)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+14, large)-Net in Base 8 — Upper bound on s

There is no (45, 59, large)-net in base 8, because

12 times m-reduction [i] would yield (45, 47, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]