MinT - Best Known (189−40, 189, s)-Nets in Base 4

Best Known (189−40, 189, s)-Nets in Base 4

(189−40, 189, 1056)-Net over F₄ — Constructive and digital

Digital (149, 189, 1056)-net over F₄, using

4¹ times duplication [i] based on digital (148, 188, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 47, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(189−40, 189, 4322)-Net over F₄ — Digital

Digital (149, 189, 4322)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁹, 4322, F₄, 40) (dual of [4322, 4133, 41]-code), using
- 206 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 56 times 0, 1, 88 times 0) [i] based on linear OA(4¹⁸², 4109, F₄, 40) (dual of [4109, 3927, 41]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(37) [i] based on
    1. linear OA(4¹⁸¹, 4096, F₄, 41) (dual of [4096, 3915, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(4¹⁶⁹, 4096, F₄, 38) (dual of [4096, 3927, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(189−40, 189, 1354087)-Net in Base 4 — Upper bound on s

There is no (149, 189, 1354088)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615661 457195 036688 564375 621872 574018 906550 364440 889978 873715 043464 804488 992335 104002 853402 719298 598394 543058 494606 > 4¹⁸⁹ [i]