MinT - Best Known (213−47, 213, s)-Nets in Base 4

Best Known (213−47, 213, s)-Nets in Base 4

(213−47, 213, 1052)-Net over F₄ — Constructive and digital

Digital (166, 213, 1052)-net over F₄, using

4¹ times duplication [i] based on digital (165, 212, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 53, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(213−47, 213, 3996)-Net over F₄ — Digital

Digital (166, 213, 3996)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹³, 3996, F₄, 47) (dual of [3996, 3783, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹³, 4110, F₄, 47) (dual of [4110, 3897, 48]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²¹², 4109, F₄, 47) (dual of [4109, 3897, 48]-code), using
    - constructionÂ X applied to C_e(46) âŠ‚ C_e(44) [i] based on
      1. linear OA(4²¹¹, 4096, F₄, 47) (dual of [4096, 3885, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
      2. linear OA(4¹⁹⁹, 4096, F₄, 45) (dual of [4096, 3897, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 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]

(213−47, 213, 1113653)-Net in Base 4 — Upper bound on s

There is no (166, 213, 1113654)-net in base 4, because

1 times m-reduction [i] would yield (166, 212, 1113654)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 322980 337272 778471 764776 268759 696792 344646 701415 894919 401883 001921 100754 504937 135370 524834 163746 757337 202986 387312 234901 348352 > 4²¹² [i]