MinT - Best Known (243−54, 243, s)-Nets in Base 4

Best Known (243−54, 243, s)-Nets in Base 4

(243−54, 243, 1052)-Net over F₄ — Constructive and digital

Digital (189, 243, 1052)-net over F₄, using

1 times m-reduction [i] based on digital (189, 244, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 61, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(243−54, 243, 4116)-Net over F₄ — Digital

Digital (189, 243, 4116)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴³, 4116, F₄, 54) (dual of [4116, 3873, 55]-code), using
- 12 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 1, 9 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]

(243−54, 243, 954586)-Net in Base 4 — Upper bound on s

There is no (189, 243, 954587)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 792990 298513 671477 459240 197927 309801 804030 191288 005414 438763 675329 352967 517082 972823 711065 867742 684957 362940 687057 474863 235608 970736 556465 319100 > 4²⁴³ [i]