MinT - Best Known (192, 192+54, s)-Nets in Base 4

Best Known (192, 192+54, s)-Nets in Base 4

(192, 192+54, 1539)-Net over F₄ — Constructive and digital

Digital (192, 246, 1539)-net over F₄, using

trace code for nets [i] based on digital (28, 82, 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]

(192, 192+54, 4310)-Net over F₄ — Digital

Digital (192, 246, 4310)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁶, 4310, F₄, 54) (dual of [4310, 4064, 55]-code), using
- 203 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 1, 9 times 0, 1, 30 times 0, 1, 65 times 0, 1, 93 times 0) [i] based on linear OA(4²⁴¹, 4102, F₄, 54) (dual of [4102, 3861, 55]-code), using
  - constructionÂ X applied to C_e(53) âŠ‚ C_e(52) [i] based on
    1. linear OA(4²⁴¹, 4096, F₄, 54) (dual of [4096, 3855, 55]-code), using an extension C_e(53) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,53], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 54 [i]
    2. linear OA(4²³⁵, 4096, F₄, 53) (dual of [4096, 3861, 54]-code), using an extension C_e(52) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,52], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 53 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(192, 192+54, 1113556)-Net in Base 4 — Upper bound on s

There is no (192, 246, 1113557)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12786 714212 722360 821520 316735 886316 714113 119672 356760 459508 100903 488686 721311 782515 405698 031197 663735 192082 266526 317990 561440 827868 495907 032075 122400 > 4²⁴⁶ [i]