MinT - Best Known (244−46, 244, s)-Nets in Base 4

Best Known (244−46, 244, s)-Nets in Base 4

(244−46, 244, 1539)-Net over F₄ — Constructive and digital

Digital (198, 244, 1539)-net over F₄, using

11 times m-reduction [i] based on 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]

(244−46, 244, 12121)-Net over F₄ — Digital

Digital (198, 244, 12121)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁴, 12121, F₄, 46) (dual of [12121, 11877, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁴⁴, 16410, F₄, 46) (dual of [16410, 16166, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(41) [i] based on
    1. linear OA(4²³⁹, 16384, F₄, 46) (dual of [16384, 16145, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    2. linear OA(4²¹⁸, 16384, F₄, 42) (dual of [16384, 16166, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    3. linear OA(4⁵, 26, F₄, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(244−46, 244, 7663112)-Net in Base 4 — Upper bound on s

There is no (198, 244, 7663113)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 799 169764 909386 985793 217470 227172 879420 865531 202118 748999 430689 666297 316736 997027 441696 761261 195122 229843 619096 506031 982157 891110 873522 621181 436160 > 4²⁴⁴ [i]