MinT - Information on Result #1749767

Information on Result #1749767

Digital (77, 87, 43697)-net over F₂, using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OOA(2⁸⁷, 43697, F₂, 3, 10) (dual of [(43697, 3), 131004, 11]-NRT-code), using

OOA 3-folding [i] based on linear OA(2⁸⁷, 131091, F₂, 10) (dual of [131091, 131004, 11]-code), using
- strength reduction [i] based on linear OA(2⁸⁷, 131091, F₂, 11) (dual of [131091, 131004, 12]-code), using
  - constructionÂ X4 applied to C_e(10) âŠ‚ C_e(8) [i] based on
    1. linear OA(2⁸⁶, 131072, F₂, 11) (dual of [131072, 130986, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 131071Â = 2¹⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(2⁶⁹, 131072, F₂, 9) (dual of [131072, 131003, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 131071Â = 2¹⁷âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(2¹⁸, 19, F₂, 17) (dual of [19, 1, 18]-code), using
      - strength reduction [i] based on linear OA(2¹⁸, 19, F₂, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
        dual of repetition code with length 19 [i]
    4. linear OA(2¹, 19, F₂, 1) (dual of [19, 18, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]

Mode: Linear.

Optimality

Show details for fixed k and m, k and s, k and t, m and s, m and t, t and s.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Digital (78, 88, 43697)-net over F₂	[i]		Net Duplication
2	Digital (79, 89, 43697)-net over F₂	[i]		Net Duplication