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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=x^2 + ux + 1 $ $ [32,24,4] $
    $ 8 $ $ g=x^4+(uv+u)x^3+(v+u)x^2+ux+1 $ $ [32,16,8] $
    $ 6 $ $ h=x^2+(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=x^5+(u+v+1)x^4+x^3+x^2+(uv+v+1)x+v+1 $ $ [48,20,12] $
    $ 12 $ $ g=x^4+(uv+u+1)x^3+x+1 $ $ [48,32,6] $
    $ 12 $ $ g=x^5+ux^4+x^3+ (u+v+ 1)x^2+v+1 $ $ [48,28,8] $
    $ 16 $ $ h=x^4+vx^3+ux^2+1 $ $ [64,24,16] $
     | Show Table
    DownLoad: CSV

    Table 3.  New binary QC codes

    n g(x) or h(x) Binary Parameters
    $ 8 $ $ g=x^3+(uv+1)x^2+x+1 $ $ [32,20,4] $
    $ 10 $ $ g=x^4+(u+v+ 1)x^3+x^2+x+1 $ $ [40,24,4] $
    $ 12 $ $ g=x^3+ux^2+1 $ $ [48,36,4] $
    $ 14 $ $ h=x^5+(uv+1)x^4+x^3+uvx^2+1 $ $ [56,20,7] $
    $ 14 $ $ h=x^4+(u+v+1)x^3+x^2+1 $ $ [56,16,12] $
    $ 14 $ $ g=x^3+ (u+v+1)x^2+1 $ $ [56,44,3] $
    $ 14 $ $ h=x^3+ (u+v+1)x^2+1 $ $ [56,12,16] $
    $ 16 $ $ h=x^6+(uv+v+1)x^4+ux^3+x^2+vx+uv+1 $ $ [64,24,16] $
    $ 20 $ $ h=x^4+x^3+(u+1)x^2+x+u+v+1 $ $ [80,16,28] $
    $ 18 $ $ h=x^4+(u+v+1)x^3+(u+v+1)x+1 $ $ [72,16,9] $
    $ 28 $ $ h=x+u+v+1 $ $ [112,4,56] $
    $ 28 $ $ h=x^4+x^2+(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=x^3+(u+1)x^2+x+1 $ $ [128,12,32] $
     | Show Table
    DownLoad: CSV

    Table 4.  New quaternary codes

    n g(x) or h(x) $ {\mathbb{Z}}_4 $ Parameters
    $ 8 $ $ g=x^3+(u+1)x^2+3x+u+1 $ $ [24,10,9] $
    $ 8 $ $ g=x^2+(3u+2)x+3u+3 $ $ [24,12,7] $
    $ 8 $ $ g=x+u+1 $ $ [16,14,2] $
    $ 12 $ $ g=x^2+(u+3)x+1 $ $ [36,20,8] $
    $ 12 $ $ h=x^4+(3u+1)x^3+(u+2)x^2+(3u+3)x+3u +3 $ $ [24,8,12] $
    $ 12 $ $ h=x^4+(3u+3)x^3+(u+2)x^2+(u+3)x+u+1 $ $ [36,8,20] $
    $ 12 $ $ h=x^5+(2u+1)x^4+3x^3+(2u+1)x^2+3x+1 $ $ [24,10,8] $
    $ 12 $ $ h=x^5+2ux^4+(3u+3)x^3+(2u+3)x^2+(3u+2)x+3 $ $ [36,10,17] $
    $ 12 $ $ g=x^5+3ux^4+x^3+(3u+1)x^2+1 $ $ [36,14,13] $
    $ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [28,8,18] $
    $ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [42,8,26] $
    $ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [28,6,18] $
    $ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [42,6,29] $
    $ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [28,20,6] $
    $ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [42,20,13] $
    $ 16 $ $ g=x^4 + (u + 2)x^3 + ux^2 + 2x + 1 $ $ [48,24,13] $
    $ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [32,26,4] $
    $ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [48,26,11] $
    $ 16 $ $ g=x^2+(u+2)x+1 $ $ [32,28,2] $
    $ 16 $ $ g=x^2+(u+2)x+1 $ $ [48,28,9] $
    $ 18 $ $ g=x^4 + (2u + 3)x^3 + ux^2 + (2u + 1)x + 3u + 3 $ $ [54,28,13] $
    $ 18 $ $ g=x^3 + ux^2 + 2x + 1 $ $ [54,30,10] $
    $ 18 $ $ h=x^3+(3u+2)x^2+3ux +2u + 3 $ $ [54,6,34] $
    $ 20 $ $ g=x^4 + 3x^3 + (3u + 1)x^2 + (3u + 1)x + 1 $ $ [60,32,13] $
    $ 20 $ $ g=x^4 + (u + 3)x^3 + x^2 + x + 1 $ $ [40,32,4] $
    $ 20 $ $ g=x^3 + (3u + 3)x^2 + (2u + 1)x + u + 3 $ $ [60,34,12] $
    $ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [48,42,4] $
    $ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [72,42,10] $
    $ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [60,52,3] $
    $ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [90,52,8] $
    $ 32 $ $ g=x + u + 1 $ $ [64,62,2] $
    $ 32 $ $ g=x + u + 1 $ $ [96,62,5] $
    $ 32 $ $ h=x^4 + ux^2 + (u + 2)x + 3u + 1 $ $ [96,8,60] $
     | Show Table
    DownLoad: CSV
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