MinT - Best Known (33−8, 33, s)-Nets in Base 8

Best Known (33−8, 33, s)-Nets in Base 8

(33−8, 33, 2050)-Net over F₈ — Constructive and digital

Digital (25, 33, 2050)-net over F₈, using

8¹ times duplication [i] based on digital (24, 32, 2050)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8³², 2050, F₈, 8, 8) (dual of [(2050, 8), 16368, 9]-NRT-code), using
  - OA 4-folding and stacking [i] based on linear OA(8³², 8200, F₈, 8) (dual of [8200, 8168, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(8³², 8202, F₈, 8) (dual of [8202, 8170, 9]-code), using
      - trace code [i] based on linear OA(64¹⁶, 4101, F₆₄, 8) (dual of [4101, 4085, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [i] based on
        linear OA(64¹⁵, 4096, F₆₄, 8) (dual of [4096, 4081, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹¹, 4096, F₆₄, 6) (dual of [4096, 4085, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(33−8, 33, 8738)-Net over F₈ — Digital

Digital (25, 33, 8738)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−8, 33, large)-Net in Base 8 — Upper bound on s

There is no (25, 33, large)-net in base 8, because

6 times m-reduction [i] would yield (25, 27, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]