August  2013, 7(3): 349-378. doi: 10.3934/amc.2013.7.349

On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

1. 

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

Received  March 2013 Published  July 2013

In [7], self-orthogonal additive codes over $\mathbb{F}_4$ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. We examine a number of classical results from the theory of $\mathbb{F}_q$-linear codes, and see how they must be modified to give analogous results for $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.
Citation: 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
References:
[1]

I. M. Araújo, et al., GAP Reference Manual,, The GAP Group, ().   Google Scholar

[2]

Report AFCRL-67-0365, Air Force Cambridge Res. Labs., Bedford, MA, 1967. Google Scholar

[3]

J. Théorie Nombres Bordeaux, 12 (2000), 225-272. doi: 10.5802/jtnb.278.  Google Scholar

[4]

in "Arithmetic of Finite Fields: First International Workshop'' (eds. C. Carlet and B. Sunar), Madrid, (2007), 276-283. doi: 10.1007/978-3-540-73074-3_21.  Google Scholar

[5]

J. Combin. Des., 8 (2000), 174-188. doi: 10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T.  Google Scholar

[6]

MacMillan Publishing, New York, 1977. Google Scholar

[7]

IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387. doi: 10.1109/18.681315.  Google Scholar

[8]

Adv. Math. Commun., 3 (2009), 329-348. doi: 10.3934/amc.2009.3.329.  Google Scholar

[9]

IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511. doi: 10.1109/TIT.2012.2196255.  Google Scholar

[10]

J. Comb. Theory Ser. A, 113 (2006), 1351-1367. doi: 10.1016/j.jcta.2005.12.004.  Google Scholar

[11]

Des. Codes Cryptogr., 34 (2005), 89-116. doi: 10.1007/s10623-003-4196-x.  Google Scholar

[12]

Marcel Dekker, New York, 1971.  Google Scholar

[13]

J. Algebra, 192 (1997), 209-234. doi: 10.1006/jabr.1996.6939.  Google Scholar

[14]

Des. Codes Cryptogr., 18 (1999), 125-148. doi: 10.1023/A:1008389220478.  Google Scholar

[15]

Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.  Google Scholar

[16]

in "Codes and Association Schemes: DIMACS Workshop'' (eds. A. Barg and S. Litsyn), Amer. Math. Soc., Providence, (2001), 135-149.  Google Scholar

[17]

Actes Congrés Internl. de Mathématique, Gauthier-Villars, Paris, (1971), 211-215.  Google Scholar

[18]

Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y.  Google Scholar

[19]

Congr. Numer., 103 (1994), 41-53.  Google Scholar

[20]

IEEE Trans. Inform. Theory, IT-49 (2003), 53-59. doi: 10.1109/TIT.2002.806146.  Google Scholar

[21]

Adv. Math. Commun., 1 (2007), 429-461. doi: 10.3934/amc.2007.1.427.  Google Scholar

[22]

Adv. Math. Commun., 2 (2008), 309-343. doi: 10.3934/amc.2008.2.309.  Google Scholar

[23]

Int. J. Inf. Coding Theory, 1 (2010), 249-284. doi: 10.1504/IJICOT.2010.032543.  Google Scholar

[24]

Adv. Math. Commun., 7 (2013), 57-90. doi: 10.3934/amc.2013.7.57.  Google Scholar

[25]

Cambridge University Press, Cambridge, 2003.  Google Scholar

[26]

Des. Codes Cryptogr., 52 (2009), 363-380. doi: 10.1007/s10623-009-9286-y.  Google Scholar

[27]

Discrete Math., 308 (2008), 3115-3124. doi: 10.1016/j.disc.2007.08.038.  Google Scholar

[28]

in "Proc. Intern. Congress Math.,'' Amer. Math. Soc., Providence, (1987), 457-465.  Google Scholar

[29]

Reading MA: WA Benjamin, 1973.  Google Scholar

[30]

Bell System Tech. J., 42 (1963), 79-94.  Google Scholar

[31]

Elsevier, New York, 1977. Google Scholar

[32]

Inform. Control, 6 (1963), 147-152. doi: 10.1016/S0019-9958(63)90189-X.  Google Scholar

[33]

IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832. doi: 10.1109/18.782103.  Google Scholar

[34]

in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.  Google Scholar

[35]

Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.  Google Scholar

[36]

Proc. Sympos. Pure Math., 34 (1979), 273-308.  Google Scholar

[37]

Heldermann Verlag, Berlin, 1992.  Google Scholar

[38]

Arch. Math. (Basel), 36 (1981), 485-494. doi: 10.1007/BF01223730.  Google Scholar

[39]

in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 827-870.  Google Scholar

show all references

References:
[1]

I. M. Araújo, et al., GAP Reference Manual,, The GAP Group, ().   Google Scholar

[2]

Report AFCRL-67-0365, Air Force Cambridge Res. Labs., Bedford, MA, 1967. Google Scholar

[3]

J. Théorie Nombres Bordeaux, 12 (2000), 225-272. doi: 10.5802/jtnb.278.  Google Scholar

[4]

in "Arithmetic of Finite Fields: First International Workshop'' (eds. C. Carlet and B. Sunar), Madrid, (2007), 276-283. doi: 10.1007/978-3-540-73074-3_21.  Google Scholar

[5]

J. Combin. Des., 8 (2000), 174-188. doi: 10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T.  Google Scholar

[6]

MacMillan Publishing, New York, 1977. Google Scholar

[7]

IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387. doi: 10.1109/18.681315.  Google Scholar

[8]

Adv. Math. Commun., 3 (2009), 329-348. doi: 10.3934/amc.2009.3.329.  Google Scholar

[9]

IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511. doi: 10.1109/TIT.2012.2196255.  Google Scholar

[10]

J. Comb. Theory Ser. A, 113 (2006), 1351-1367. doi: 10.1016/j.jcta.2005.12.004.  Google Scholar

[11]

Des. Codes Cryptogr., 34 (2005), 89-116. doi: 10.1007/s10623-003-4196-x.  Google Scholar

[12]

Marcel Dekker, New York, 1971.  Google Scholar

[13]

J. Algebra, 192 (1997), 209-234. doi: 10.1006/jabr.1996.6939.  Google Scholar

[14]

Des. Codes Cryptogr., 18 (1999), 125-148. doi: 10.1023/A:1008389220478.  Google Scholar

[15]

Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.  Google Scholar

[16]

in "Codes and Association Schemes: DIMACS Workshop'' (eds. A. Barg and S. Litsyn), Amer. Math. Soc., Providence, (2001), 135-149.  Google Scholar

[17]

Actes Congrés Internl. de Mathématique, Gauthier-Villars, Paris, (1971), 211-215.  Google Scholar

[18]

Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y.  Google Scholar

[19]

Congr. Numer., 103 (1994), 41-53.  Google Scholar

[20]

IEEE Trans. Inform. Theory, IT-49 (2003), 53-59. doi: 10.1109/TIT.2002.806146.  Google Scholar

[21]

Adv. Math. Commun., 1 (2007), 429-461. doi: 10.3934/amc.2007.1.427.  Google Scholar

[22]

Adv. Math. Commun., 2 (2008), 309-343. doi: 10.3934/amc.2008.2.309.  Google Scholar

[23]

Int. J. Inf. Coding Theory, 1 (2010), 249-284. doi: 10.1504/IJICOT.2010.032543.  Google Scholar

[24]

Adv. Math. Commun., 7 (2013), 57-90. doi: 10.3934/amc.2013.7.57.  Google Scholar

[25]

Cambridge University Press, Cambridge, 2003.  Google Scholar

[26]

Des. Codes Cryptogr., 52 (2009), 363-380. doi: 10.1007/s10623-009-9286-y.  Google Scholar

[27]

Discrete Math., 308 (2008), 3115-3124. doi: 10.1016/j.disc.2007.08.038.  Google Scholar

[28]

in "Proc. Intern. Congress Math.,'' Amer. Math. Soc., Providence, (1987), 457-465.  Google Scholar

[29]

Reading MA: WA Benjamin, 1973.  Google Scholar

[30]

Bell System Tech. J., 42 (1963), 79-94.  Google Scholar

[31]

Elsevier, New York, 1977. Google Scholar

[32]

Inform. Control, 6 (1963), 147-152. doi: 10.1016/S0019-9958(63)90189-X.  Google Scholar

[33]

IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832. doi: 10.1109/18.782103.  Google Scholar

[34]

in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.  Google Scholar

[35]

Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.  Google Scholar

[36]

Proc. Sympos. Pure Math., 34 (1979), 273-308.  Google Scholar

[37]

Heldermann Verlag, Berlin, 1992.  Google Scholar

[38]

Arch. Math. (Basel), 36 (1981), 485-494. doi: 10.1007/BF01223730.  Google Scholar

[39]

in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 827-870.  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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006

[15]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[16]

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

[17]

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

[18]

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

[19]

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

[20]

Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]