MinT - Best Known (184, 184+50, s)-Nets in Base 4

Best Known (184, 184+50, s)-Nets in Base 4

(184, 184+50, 1539)-Net over F₄ — Constructive and digital

Digital (184, 234, 1539)-net over F₄, using

trace code for nets [i] based on digital (28, 78, 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]

(184, 184+50, 4804)-Net over F₄ — Digital

Digital (184, 234, 4804)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁴, 4804, F₄, 50) (dual of [4804, 4570, 51]-code), using
- 691 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 35 times 0, 1, 58 times 0, 1, 83 times 0, 1, 104 times 0, 1, 118 times 0, 1, 125 times 0, 1, 131 times 0) [i] based on linear OA(4²²³, 4102, F₄, 50) (dual of [4102, 3879, 51]-code), using
  - constructionÂ X applied to C_e(49) âŠ‚ C_e(48) [i] based on
    1. linear OA(4²²³, 4096, F₄, 50) (dual of [4096, 3873, 51]-code), using an extension C_e(49) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,49], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [i]
    2. linear OA(4²¹⁷, 4096, F₄, 49) (dual of [4096, 3879, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [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]

(184, 184+50, 1464805)-Net in Base 4 — Upper bound on s

There is no (184, 234, 1464806)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 147722 810617 872818 565655 302803 436266 534560 791077 951305 246932 596164 514975 099416 839647 935118 295038 481514 144465 775511 718399 437412 435158 657700 > 4²³⁴ [i]