MinT - Best Known (198, 255, s)-Nets in Base 4

Best Known (198, 255, s)-Nets in Base 4

(198, 255, 1539)-Net over F₄ — Constructive and digital

Digital (198, 255, 1539)-net over F₄, using

trace code for nets [i] based on digital (28, 85, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(198, 255, 4119)-Net over F₄ — Digital

Digital (198, 255, 4119)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁵, 4119, F₄, 57) (dual of [4119, 3864, 58]-code), using
- 15 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 14 times 0) [i] based on linear OA(4²⁵⁴, 4103, F₄, 57) (dual of [4103, 3849, 58]-code), using
  - constructionÂ X applied to C_e(56) âŠ‚ C_e(54) [i] based on
    1. linear OA(4²⁵³, 4096, F₄, 57) (dual of [4096, 3843, 58]-code), using an extension C_e(56) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,56], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 57 [i]
    2. linear OA(4²⁴⁷, 4096, F₄, 55) (dual of [4096, 3849, 56]-code), using an extension C_e(54) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,54], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 55 [i]
    3. linear OA(4¹, 7, F₄, 1) (dual of [7, 6, 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]

(198, 255, 1089980)-Net in Base 4 — Upper bound on s

There is no (198, 255, 1089981)-net in base 4, because

1 times m-reduction [i] would yield (198, 254, 1089981)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 837 990346 071484 650747 255908 967328 847449 516440 853055 398213 265036 000128 224527 149865 261466 604279 189769 843419 364608 052435 564033 883664 652905 804283 045870 717179 > 4²⁵⁴ [i]