MinT - Best Known (227−49, 227, s)-Nets in Base 4

Best Known (227−49, 227, s)-Nets in Base 4

(227−49, 227, 1056)-Net over F₄ — Constructive and digital

Digital (178, 227, 1056)-net over F₄, using

1 times m-reduction [i] based on digital (178, 228, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 57, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(227−49, 227, 4430)-Net over F₄ — Digital

Digital (178, 227, 4430)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁷, 4430, F₄, 49) (dual of [4430, 4203, 50]-code), using
- 318 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 57 times 0, 1, 82 times 0, 1, 105 times 0) [i] based on linear OA(4²¹⁸, 4103, F₄, 49) (dual of [4103, 3885, 50]-code), using
  - constructionÂ X applied to C_e(48) âŠ‚ C_e(46) [i] based on
    1. 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]
    2. 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]
    3. linear OA(4¹, 7, F₄, 1) (dual of [7, 6, 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]

(227−49, 227, 1526279)-Net in Base 4 — Upper bound on s

There is no (178, 227, 1526280)-net in base 4, because

1 times m-reduction [i] would yield (178, 226, 1526280)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11629 438042 795935 830968 766570 568096 274996 316014 369838 836073 088784 444956 333856 784746 626504 839077 281319 635495 438677 421291 550830 009749 630656 > 4²²⁶ [i]