MinT - Best Known (67−22, 67, s)-Nets in Base 4

Best Known (67−22, 67, s)-Nets in Base 4

(67−22, 67, 195)-Net over F₄ — Constructive and digital

Digital (45, 67, 195)-net over F₄, using

4¹ times duplication [i] based on digital (44, 66, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 22, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(67−22, 67, 255)-Net over F₄ — Digital

Digital (45, 67, 255)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶⁷, 255, F₄, 22) (dual of [255, 188, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶⁷, 272, F₄, 22) (dual of [272, 205, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(17) [i] based on
    1. linear OA(4⁶³, 256, F₄, 22) (dual of [256, 193, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(4⁵¹, 256, F₄, 18) (dual of [256, 205, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(4⁴, 16, F₄, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]

(67−22, 67, 7594)-Net in Base 4 — Upper bound on s

There is no (45, 67, 7595)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21798 641044 491814 029745 923583 745888 009500 > 4⁶⁷ [i]