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Skew constacyclic codes over the local Frobenius non-chain rings of order 16

  • * Corresponding author: Steven T. Dougherty

    * Corresponding author: Steven T. Dougherty 

Esengül Saltürk would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper

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  • We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

    Mathematics Subject Classification: Primary: 11T71; Secondary: 94B15, 13H99.

    Citation:

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  • Table 1.  Optimal binary linear codes

    n g(x) or h(x) Binary Parameters
    8 g=x2+ux+1 [32,24,4]
    8 g=x4+(uv+u)x3+(v+u)x2+ux+1 [32,16,8]
    6 h=x2+(u+v+1)x+1 [24,8,8]
    6 h=x+uv+1 [24,4,12]
     | Show Table
    DownLoad: CSV

    Table 2.  Best-known binary linear codes

    n g(x) or h(x) Binary Parameters
    12 h=x5+(u+v+1)x4+x3+x2+(uv+v+1)x+v+1 [48,20,12]
    12 g=x4+(uv+u+1)x3+x+1 [48,32,6]
    12 g=x5+ux4+x3+(u+v+1)x2+v+1 [48,28,8]
    16 h=x4+vx3+ux2+1 [64,24,16]
     | Show Table
    DownLoad: CSV

    Table 3.  New binary QC codes

    n g(x) or h(x) Binary Parameters
    8 g=x3+(uv+1)x2+x+1 [32,20,4]
    10 g=x4+(u+v+1)x3+x2+x+1 [40,24,4]
    12 g=x3+ux2+1 [48,36,4]
    14 h=x5+(uv+1)x4+x3+uvx2+1 [56,20,7]
    14 h=x4+(u+v+1)x3+x2+1 [56,16,12]
    14 g=x3+(u+v+1)x2+1 [56,44,3]
    14 h=x3+(u+v+1)x2+1 [56,12,16]
    16 h=x6+(uv+v+1)x4+ux3+x2+vx+uv+1 [64,24,16]
    20 h=x4+x3+(u+1)x2+x+u+v+1 [80,16,28]
    18 h=x4+(u+v+1)x3+(u+v+1)x+1 [72,16,9]
    28 h=x+u+v+1 [112,4,56]
    28 h=x4+x2+(u+1)x+(u+1)v+1 [112,16,40]
    30 h=x+uv+1 [120,4,60]
    32 h=x+u+v+1 [128,4,64]
    32 h=x3+(u+1)x2+x+1 [128,12,32]
     | Show Table
    DownLoad: CSV

    Table 4.  New quaternary codes

    n g(x) or h(x) Z4 Parameters
    8 g=x3+(u+1)x2+3x+u+1 [24,10,9]
    8 g=x2+(3u+2)x+3u+3 [24,12,7]
    8 g=x+u+1 [16,14,2]
    12 g=x2+(u+3)x+1 [36,20,8]
    12 h=x4+(3u+1)x3+(u+2)x2+(3u+3)x+3u+3 [24,8,12]
    12 h=x4+(3u+3)x3+(u+2)x2+(u+3)x+u+1 [36,8,20]
    12 h=x5+(2u+1)x4+3x3+(2u+1)x2+3x+1 [24,10,8]
    12 h=x5+2ux4+(3u+3)x3+(2u+3)x2+(3u+2)x+3 [36,10,17]
    12 g=x5+3ux4+x3+(3u+1)x2+1 [36,14,13]
    14 h=x4+(3u+3)x3+(u+3)x2+ux+u+1 [28,8,18]
    14 h=x4+(3u+3)x3+(u+3)x2+ux+u+1 [42,8,26]
    14 h=x3+(u+3)x2+(3u+2)x+u+1 [28,6,18]
    14 h=x3+(u+3)x2+(3u+2)x+u+1 [42,6,29]
    14 g=x4+(u+3)x3+x2+(3u+2)x+2u+1 [28,20,6]
    14 g=x4+(u+3)x3+x2+(3u+2)x+2u+1 [42,20,13]
    16 g=x4+(u+2)x3+ux2+2x+1 [48,24,13]
    16 g=x3+(u+3)x2+(3u+1)x+3u+3 [32,26,4]
    16 g=x3+(u+3)x2+(3u+1)x+3u+3 [48,26,11]
    16 g=x2+(u+2)x+1 [32,28,2]
    16 g=x2+(u+2)x+1 [48,28,9]
    18 g=x4+(2u+3)x3+ux2+(2u+1)x+3u+3 [54,28,13]
    18 g=x3+ux2+2x+1 [54,30,10]
    18 h=x3+(3u+2)x2+3ux+2u+3 [54,6,34]
    20 g=x4+3x3+(3u+1)x2+(3u+1)x+1 [60,32,13]
    20 g=x4+(u+3)x3+x2+x+1 [40,32,4]
    20 g=x3+(3u+3)x2+(2u+1)x+u+3 [60,34,12]
    24 g=x3+3ux2+(u+2)x+1 [48,42,4]
    24 g=x3+3ux2+(u+2)x+1 [72,42,10]
    30 g=x4+(u+1)x3+1 [60,52,3]
    30 g=x4+(u+1)x3+1 [90,52,8]
    32 g=x+u+1 [64,62,2]
    32 g=x+u+1 [96,62,5]
    32 h=x4+ux2+(u+2)x+3u+1 [96,8,60]
     | Show Table
    DownLoad: CSV
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