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Article Contents

# An overview on skew constacyclic codes and their subclass of LCD codes

The third author is supported by the French government Investissements d'Avenir program ANR-11-LABX-0020-01
• This paper is about a first characterization of LCD skew constacyclic codes and some constructions of LCD skew cyclic and skew negacyclic codes over $\mathbb{F}_{p^2}$.

Mathematics Subject Classification: 94B05, 12-08, 68W30, 12E15.

 Citation:

• Table 1.  Number of Euclidean and Hermitian LCD $[2p,p]_{p^2}$ and $[2p,p]_{p^2}$ MDS skew-cyclic codes for $p = 3,5,7,11$

 p nbr of Euclidean LCD skew cyc. nbr of Hermitian LCD skew cyc. $[2p,p]_{p^2}$ $[2p,p,p+1]_{p^2}$ $[2p,p]_{p^2}$ $[2p,p,p+1]_{p^2}$ 3 18 16 36 32 5 3750 2412 3750 2412 7 705984 39564 941192 52752 11 259374246010 $\geq 1$ 311249095212 $\geq 1$

Table 5.  Dimensions of MDS LCD skew codes over ${\mathbb F}_9$ with length $n \leq 10$ and dimension $1 < k < n-1$

 MDS Euclidean LCD MDS Hermitian LCD length skew cyc skew nega skew cyc skew nega 4 2 2 no 2 6 3 3 3 no 8 3, 4, 5 4 3, 5 4 10 5 5 5 no

Table 6.  Dimensions of MDS LCD skew codes over ${\mathbb F}_{25}$ with length $n \leq 18$ and dimension $1 < k < n-1$

 MDS Euclidean LCD MDS Hermitian LCD length skew cyc skew nega skew cyc skew nega 4 2 2 no 2 6 2, 3, 4 2, 3, 4 2, 3, 4 2, 4 8 3, 4, 5 4 3, 5 4 10 5 5 5 no 12 3, 5, 6, 7, 9 6 3, 5, 7, 9 6 14 7 7 7 no 16 7, 8, 9 no 7, 9 no 18 9 9 9 no

Table 7.  Dimensions of MDS LCD skew codes over ${\mathbb F}_{49}$ with length $n \leq 16$ and dimension $1 < k < n-1$

 MDS Euclidean LCD MDS Hermitian LCD length skew cyc skew nega skew cyc skew nega 4 2 2 no 2 6 2, 3, 4 2, 3, 4 2, 3, 4 2, 4 8 3, 4, 5 2, 4, 6 3, 5 2, 4, 6 10 4, 5, 6 4, 5, 6 4, 5, 6 4, 6 12 3, 5, 6, 7, 9 6 3, 5, 7, 9 6 14 7 7 7 no 16 3, 5, 7, 8, 9, 11, 13 8 3, 5, 7, 9, 11, 13 8

Table 2.  Best minimum distances and numbers of Euclidean LCD $[2k,k]$ skew cyclic codes of length $\leq 48$ over ${\mathbb F}_4$ with skew generator polynomial not divisible by a central polynomial

 Euclidean LCD skew cyc. Euclidean LCD skew cyc. length best dist nbr length best dist nbr 2 2* 2 26 9 8 064 4 3* 4 28 11* 18 432 6 4* 4 30 12* 13 056 8 4* 16 32 10 65 536 10 5* 24 34 11* 115 200 12 5 32 36 11 114 688 14 6* 144 38 12* 523 264 16 6 256 40 12* 786 432 18 7 224 42 12 1 198 080 20 8* 768 44 13 4 063 232 22 8* 1 984 46 14* 8 392 704 24 9* 2 048 48 14* 8 388 608

Table 3.  Best minimum distances and numbers of LCD $[2k,k]$ skew codes of length $\leq 24$ over ${\mathbb F}_9$ with skew generator polynomial not divisible by a central polynomial

 Euclidean LCD skew cyc skew negacyc length best dist nbr best dist nbr 2 2* 2 2* 4 4 3* 32 3* 6 6 4* 18 4* 36 8 5* 192 5* 90 10 6* 144 6* 288 12 6* 5 408 6* 486 14 7* 1 404 7* 2 808 16 7 17 280 8* 6 642 18 9* 13 122 9* 26 244 20 9 165 888 9 39 852 22 9* 118 584 9* 237 168 24 10* 2 628 288 10* 590 490

Table 4.  Best minimum distances and numbers of LCD $[2k,k]$ skew codes of length $\leq 24$ over ${\mathbb F}_9$ with skew generator polynomial not divisible by a central polynomial

 Hermitian LCD skew cyc skew negacyc length best dist nbr best dist nbr 2 2* 4 0 0 4 0 0 3* 6 6 4* 361 0 0 8 0 0 5* 90 10 6* 288 0 0 12 0 0 6* 486 14 7* 2 808 0 0 16 0 0 8* 6 642 18 9* 26 244 0 0 20 0 0 10* 39 852 22 9* 237 168 0 0 24 0 0 10* 590 490
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