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Article Contents

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

• * 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

• 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:

• Table 1.  Optimal binary linear codes

 n g(x) or h(x) Binary Parameters $8$ $g=x^2 + ux + 1$ $[32,24,4]$ $8$ $g=x^4+(uv+u)x^3+(v+u)x^2+ux+1$ $[32,16,8]$ $6$ $h=x^2+(u+v+1)x + 1$ $[24,8,8]$ $6$ $h=x+uv+1$ $[24,4,12]$

Table 2.  Best-known binary linear codes

 n g(x) or h(x) Binary Parameters $12$ $h=x^5+(u+v+1)x^4+x^3+x^2+(uv+v+1)x+v+1$ $[48,20,12]$ $12$ $g=x^4+(uv+u+1)x^3+x+1$ $[48,32,6]$ $12$ $g=x^5+ux^4+x^3+ (u+v+ 1)x^2+v+1$ $[48,28,8]$ $16$ $h=x^4+vx^3+ux^2+1$ $[64,24,16]$

Table 3.  New binary QC codes

 n g(x) or h(x) Binary Parameters $8$ $g=x^3+(uv+1)x^2+x+1$ $[32,20,4]$ $10$ $g=x^4+(u+v+ 1)x^3+x^2+x+1$ $[40,24,4]$ $12$ $g=x^3+ux^2+1$ $[48,36,4]$ $14$ $h=x^5+(uv+1)x^4+x^3+uvx^2+1$ $[56,20,7]$ $14$ $h=x^4+(u+v+1)x^3+x^2+1$ $[56,16,12]$ $14$ $g=x^3+ (u+v+1)x^2+1$ $[56,44,3]$ $14$ $h=x^3+ (u+v+1)x^2+1$ $[56,12,16]$ $16$ $h=x^6+(uv+v+1)x^4+ux^3+x^2+vx+uv+1$ $[64,24,16]$ $20$ $h=x^4+x^3+(u+1)x^2+x+u+v+1$ $[80,16,28]$ $18$ $h=x^4+(u+v+1)x^3+(u+v+1)x+1$ $[72,16,9]$ $28$ $h=x+u+v+1$ $[112,4,56]$ $28$ $h=x^4+x^2+(u+1)x+(u+1)v+1$ $[112,16,40]$ $30$ $h= x+uv+1$ $[120,4,60]$ $32$ $h=x+u+v+1$ $[128,4,64]$ $32$ $h=x^3+(u+1)x^2+x+1$ $[128,12,32]$

Table 4.  New quaternary codes

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