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February  2013, 7(1): 57-90. doi: 10.3934/amc.2013.7.57

Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order

W. Cary Huffman 1,

1. 

Department of Mathematics and Statistics, Loyola University, Chicago, IL 60660, United States

Received  June 2012 Revised  August 2012 Published  January 2013

Additive codes over $\mathbb{F}_4$ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic $p$ are connected to codes over $\mathbb{F}_{p^2}$ that are $\mathbb{F}_p$-linear in [30]. These codes that arise as connections with quantum codes are self-orthogonal under a particular inner product. We study a further generalization to codes termed $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are self-dual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case $q=3$ and $t=2$.
Keywords: Additive codes, self-dual codes, code automorphisms..
Mathematics Subject Classification: 94B1.
Citation: W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57
References:
[1]

I. M. Araújo, et al., GAP reference manual,, The GAP Group, ().   Google Scholar

[2]

in "Arithmetic of Finite Fields: First International Workshop'' (eds. C. Carlet and B. Sunar), Madrid, (2007), 276-283.  Google Scholar

[3]

J. Combin. Des., 8 (2000), 174-188.  Google Scholar

[4]

Des. Codes Crypt., 28 (2003), 163-169. doi: 10.1023/A:1022588407585.  Google Scholar

[5]

IEEE Trans. Inform. Theory, IT-56 (2010), 2091-2096.  Google Scholar

[6]

Finite Fields Appl., 13 (2007), 605-615. doi: 10.1016/j.ffa.2006.01.001.  Google Scholar

[7]

Int. J. Inform. Coding Theory, 2 (2011), 21-37.  Google Scholar

[8]

Des. Codes Crypt., 9 (1996), 131-141.  Google Scholar

[9]

IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387.  Google Scholar

[10]

Adv. Math. Commun., 3 (2009), 329-348.  Google Scholar

[11]

IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511.  Google Scholar

[12]

Des. Codes Crypt., 34 (2005), 89-116.  Google Scholar

[13]

Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/S1071-5797(02)00018-7.  Google Scholar

[14]

Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.  Google Scholar

[15]

IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.  Google Scholar

[16]

IEEE Trans. Inform. Theory, IT-36 (1990), 651-660.  Google Scholar

[17]

IEEE Trans. Inform. Theory, IT-37 (1991), 1206-1216.  Google Scholar

[18]

IEEE Trans. Inform. Theory, IT-38 (1992), 1395-1400.  Google Scholar

[19]

IEEE Trans. Inform. Theory, IT-44 (1998), 800-809.  Google Scholar

[20]

Adv. Math. Commun., 1 (2007), 357-398.  Google Scholar

[21]

Adv. Math. Commun., 1 (2007), 429-461.  Google Scholar

[22]

Finite Fields Appl., 13 (2007), 681-712. doi: 10.1016/j.ffa.2006.02.003.  Google Scholar

[23]

Adv. Math. Commun., 2 (2008), 309-343.  Google Scholar

[24]

Int. J. Inform. Coding Theory, 1 (2010), 249-284.  Google Scholar

[25]

Des. Codes Crypt., 6 (1995), 97-106. doi: 10.1007/BF01398008.  Google Scholar

[26]

Finite Fields Appl., 7 (2001), 341-349. doi: 10.1006/ffta.2000.0295.  Google Scholar

[27]

Problems Inform. Trans., 19 (1983), 260-270.  Google Scholar

[28]

Problems Inform. Trans., 22 (1986), 277-284.  Google Scholar

[29]

Discrete Math., 308 (2008), 3115-3124.  Google Scholar

[30]

IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832.  Google Scholar

[31]

Finite Fields Appl., 8 (2002), 34-51. doi: 10.1006/ffta.2001.0322.  Google Scholar

[32]

IEEE Trans. Inform. Theory, IT-57 (2011), 7498-7506.  Google Scholar

[33]

IEEE Trans. Inform. Theory, IT-33 (1987), 77-82.  Google Scholar

[34]

Discrete Math., 190 (1998), 201-213.  Google Scholar

[35]

Problems Inform. Trans., 32 (1996), 253-257.  Google Scholar

show all references

References:
[1]

I. M. Araújo, et al., GAP reference manual,, The GAP Group, ().   Google Scholar

[2]

in "Arithmetic of Finite Fields: First International Workshop'' (eds. C. Carlet and B. Sunar), Madrid, (2007), 276-283.  Google Scholar

[3]

J. Combin. Des., 8 (2000), 174-188.  Google Scholar

[4]

Des. Codes Crypt., 28 (2003), 163-169. doi: 10.1023/A:1022588407585.  Google Scholar

[5]

IEEE Trans. Inform. Theory, IT-56 (2010), 2091-2096.  Google Scholar

[6]

Finite Fields Appl., 13 (2007), 605-615. doi: 10.1016/j.ffa.2006.01.001.  Google Scholar

[7]

Int. J. Inform. Coding Theory, 2 (2011), 21-37.  Google Scholar

[8]

Des. Codes Crypt., 9 (1996), 131-141.  Google Scholar

[9]

IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387.  Google Scholar

[10]

Adv. Math. Commun., 3 (2009), 329-348.  Google Scholar

[11]

IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511.  Google Scholar

[12]

Des. Codes Crypt., 34 (2005), 89-116.  Google Scholar

[13]

Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/S1071-5797(02)00018-7.  Google Scholar

[14]

Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.  Google Scholar

[15]

IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.  Google Scholar

[16]

IEEE Trans. Inform. Theory, IT-36 (1990), 651-660.  Google Scholar

[17]

IEEE Trans. Inform. Theory, IT-37 (1991), 1206-1216.  Google Scholar

[18]

IEEE Trans. Inform. Theory, IT-38 (1992), 1395-1400.  Google Scholar

[19]

IEEE Trans. Inform. Theory, IT-44 (1998), 800-809.  Google Scholar

[20]

Adv. Math. Commun., 1 (2007), 357-398.  Google Scholar

[21]

Adv. Math. Commun., 1 (2007), 429-461.  Google Scholar

[22]

Finite Fields Appl., 13 (2007), 681-712. doi: 10.1016/j.ffa.2006.02.003.  Google Scholar

[23]

Adv. Math. Commun., 2 (2008), 309-343.  Google Scholar

[24]

Int. J. Inform. Coding Theory, 1 (2010), 249-284.  Google Scholar

[25]

Des. Codes Crypt., 6 (1995), 97-106. doi: 10.1007/BF01398008.  Google Scholar

[26]

Finite Fields Appl., 7 (2001), 341-349. doi: 10.1006/ffta.2000.0295.  Google Scholar

[27]

Problems Inform. Trans., 19 (1983), 260-270.  Google Scholar

[28]

Problems Inform. Trans., 22 (1986), 277-284.  Google Scholar

[29]

Discrete Math., 308 (2008), 3115-3124.  Google Scholar

[30]

IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832.  Google Scholar

[31]

Finite Fields Appl., 8 (2002), 34-51. doi: 10.1006/ffta.2001.0322.  Google Scholar

[32]

IEEE Trans. Inform. Theory, IT-57 (2011), 7498-7506.  Google Scholar

[33]

IEEE Trans. Inform. Theory, IT-33 (1987), 77-82.  Google Scholar

[34]

Discrete Math., 190 (1998), 201-213.  Google Scholar

[35]

Problems Inform. Trans., 32 (1996), 253-257.  Google Scholar

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