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Some shortened codes from linear codes constructed by defining sets

  • * Corresponding author: Chunming Tang

    * Corresponding author: Chunming Tang 
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  • The puncturing and shortening technologies are two important approaches to construct new linear codes. It is noticed that many works on punctured linear codes have been done in the past few decades, but till now little research on the shortening technique has been done and there are only a handful references on shortened linear codes. In this paper, we use the shortening technology to obtain some shortened codes from linear codes constructed by certain special defining sets, and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.

    Mathematics Subject Classification: Primary: 94A05; Secondary: 94B60.

    Citation:

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  • Table 1.  The weight distribution of C(˜D)

    weight multiplicity
    0 1
    (nf+12(m2)/2)/2 2m+1+(nf+1)2(2m)/22
    (nf+1+2(m2)/2)/2 2m3(nf+1)2(2m)/22
     | Show Table
    DownLoad: CSV

    Table 2.  The weight distribution of CT

    weight multiplicity
    0 1
    (nf+12(m2)/2)/2 2m11+2m/2+1(nf+1)2(2m)/22
    (nf+1+2(m2)/2)/2 2m112m/2+1+(nf+1)2(2m)/22
     | Show Table
    DownLoad: CSV

    Table 3.  The weight distribution of CT for λ=0

    weight multiplicity
    0 1
    (nf+12(m2)/2)/2 32m/211+2m2(nf+1)2(2m)/22
    (nf+1+2(m2)/2)/2 32m/211+2m2+(nf+1)2(2m)/22
     | Show Table
    DownLoad: CSV

    Table 4.  The weight distribution of CT for λ=1

    weight multiplicity
    0 1
    (nf+12(m2)/2)/2 2m/2+11+2m2(nf+1)2(2m)/22
    (nf+1+2(m2)/2)/2 2m/2+11+2m2+(nf+1)2(2m)/22
     | Show Table
    DownLoad: CSV

    Table 5.  The weight distribution of CT for Tr(α(x21x2+x22x1))=0

    weight multiplicity
    0 1
    2m22(m2)/2 2m/21+2m3
    2m2 1+2m32m/21
     | Show Table
    DownLoad: CSV

    Table 6.  The weight distribution of CT for Tr(α(x21x2+x22x1))=1

    weight multiplicity
    0 1
    2m22(m2)/2 2m/22+2m3
    2m2 1+2m32m/22
     | Show Table
    DownLoad: CSV

    Table 7.  The weight distribution of C(˘D)

    weight multiplicity
    0 1
    pm2 pm1(1)(p1)24m2pm22(p1)1
    pm2(1)(p1)24m2pm22 (p1)(pm1+(1)(p1)24m2pm22)
     | Show Table
    DownLoad: CSV

    Table 8.  The weight distribution of CT in Theorem 4.2

    weight multiplicity
    0 1
    pm2 pm2(1)(p1)24m2pm22(p1)1
    pm2(1)(p1)24m2pm22 (p1)(pm2+(1)(p1)24m2pm22)
     | Show Table
    DownLoad: CSV

    Table 9.  The weight distribution of CT for Tr(x1x2)=0

    weight multiplicity
    0 1
    pm2 pm3(1)(p1)24m2pm22(p1)1
    pm2(1)(p1)24m2pm22 (p1)(pm3+(1)(p1)24m2pm22)
     | Show Table
    DownLoad: CSV

    Table 10.  The weight distribution of CT for Tr(x1x2)0

    weight multiplicity
    0 1
    pm2 pm3(1)(p1)24m2pm42(p1)1
    pm2(1)(p1)24m2pm22 (p1)(pm3+(1)(p1)24m2pm42)
     | Show Table
    DownLoad: CSV
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