MinT - Best Known (22−7, 22, s)-Nets in Base 4

Best Known (22−7, 22, s)-Nets in Base 4

(22−7, 22, 195)-Net over F₄ — Constructive and digital

Digital (15, 22, 195)-net over F₄, using

4¹ times duplication [i] based on digital (14, 21, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 7, 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]

(22−7, 22, 266)-Net over F₄ — Digital

Digital (15, 22, 266)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²², 266, F₄, 7) (dual of [266, 244, 8]-code), using
- constructionÂ X4 applied to C_e(6) âŠ‚ C_e(4) [i] based on
  1. linear OA(4²¹, 256, F₄, 7) (dual of [256, 235, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. linear OA(4¹³, 256, F₄, 5) (dual of [256, 243, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  3. linear OA(4⁹, 10, F₄, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
    - dual of repetition code with length 10 [i]
  4. linear OA(4¹, 10, F₄, 1) (dual of [10, 9, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(22−7, 22, 9921)-Net in Base 4 — Upper bound on s

There is no (15, 22, 9922)-net in base 4, because

1 times m-reduction [i] would yield (15, 21, 9922)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4 398177 561328 > 4²¹ [i]