MinT - Information on Result #1194820

Information on Result #1194820

Digital (17, 24, 2055)-net over F₁₆, using net defined by OOA based on linear OOA(16²⁴, 2055, F₁₆, 10, 7) (dual of [(2055, 10), 20526, 8]-NRT-code), using

OOA 2-folding and stacking with additional row [i] based on linear OOA(16²⁴, 4111, F₁₆, 2, 7) (dual of [(4111, 2), 8198, 8]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16²⁴, 4112, F₁₆, 2, 7) (dual of [(4112, 2), 8200, 8]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁰, s, F₁₆, 2, 0) for arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    3. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    4. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    5. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    6. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    7. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    8. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    9. linear OOA(16⁰, 257, F₁₆, 2, 0) (dual of [(257, 2), 514, 1]-NRT-code) (see above)
    10. linear OOA(16¹, 257, F₁₆, 2, 1) (dual of [(257, 2), 513, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16¹, s, F₁₆, 2, 1) for arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
    11. linear OOA(16¹, 257, F₁₆, 2, 1) (dual of [(257, 2), 513, 2]-NRT-code) (see above)
    12. linear OOA(16¹, 257, F₁₆, 2, 1) (dual of [(257, 2), 513, 2]-NRT-code) (see above)
    13. linear OOA(16¹, 257, F₁₆, 2, 1) (dual of [(257, 2), 513, 2]-NRT-code) (see above)
    14. linear OOA(16³, 257, F₁₆, 2, 2) (dual of [(257, 2), 511, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16³, 273, F₁₆, 2, 2) (dual of [(273, 2), 543, 3]-NRT-code), using
        appending kth column [i] based on linear OA(16³, 273, F₁₆, 2) (dual of [273, 270, 3]-code), using
        Hamming code H(3,16) [i]
    15. linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
        ovoid in PG(3,Â 16) [i]
    16. linear OOA(16¹³, 257, F₁₆, 2, 7) (dual of [(257, 2), 501, 8]-NRT-code), using
      - extracting embedded OOA [i] based on digital (6, 13, 257)-net over F₁₆, using
        base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

Mode: Constructive and 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 (90, 104, 2398797)-net over F₁₆	[i]		(u,Â u+v)-Construction for Nets
2	Digital (95, 110, 2398797)-net over F₁₆	[i]		(u,Â u+v)-Construction for Nets