MinT - Best Known (77−16, 77, s)-Nets in Base 8

Best Known (77−16, 77, s)-Nets in Base 8

(77−16, 77, 4099)-Net over F₈ — Constructive and digital

Digital (61, 77, 4099)-net over F₈, using

t-expansion [i] based on digital (60, 77, 4099)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁷⁷, 4099, F₈, 17, 17) (dual of [(4099, 17), 69606, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(8⁷⁷, 32793, F₈, 17) (dual of [32793, 32716, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁷⁷, 32794, F₈, 17) (dual of [32794, 32717, 18]-code), using
      - constructionÂ X applied to C_e(16) âŠ‚ C_e(11) [i] based on
        linear OA(8⁷¹, 32768, F₈, 17) (dual of [32768, 32697, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(8⁵¹, 32768, F₈, 12) (dual of [32768, 32717, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(8⁶, 26, F₈, 4) (dual of [26, 20, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁶, 56, F₈, 4) (dual of [56, 50, 5]-code), using
        1 times truncation [i] based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]

(77−16, 77, 39684)-Net over F₈ — Digital

Digital (61, 77, 39684)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−16, 77, large)-Net in Base 8 — Upper bound on s

There is no (61, 77, large)-net in base 8, because

14 times m-reduction [i] would yield (61, 63, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]