American Institute of Mathematical Sciences

Advanced
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

[1]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077
[2]

Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese. Combining subspace codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021007
[3]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[4]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[5]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[6]

Emily McMillon, Allison Beemer, Christine A. Kelley. Extremal absorbing sets in low-density parity-check codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021003
[7]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002
[8]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[9]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135
[10]

Muhammad Ajmal, Xiande Zhang. New optimal error-correcting codes for crosstalk avoidance in on-chip data buses. Advances in Mathematics of Communications, 2021, 15 (3) : 487-506. doi: 10.3934/amc.2020078
[11]

Yun Gao, Shilin Yang, Fang-Wei Fu. Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes. Advances in Mathematics of Communications, 2021, 15 (3) : 387-396. doi: 10.3934/amc.2020072
[12]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[13]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079
[14]

Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003
[15]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074
[16]

Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075
[17]

Jinsen Guo, Yongwu Zhou, Baixun Li. The optimal pricing and service strategies of a dual-channel retailer under free riding. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021056
[18]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001
[19]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207
[20]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (164)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences