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, http://www.gap-system.org

[2]

E. F. Assmus, Jr., H. F. Mattson, Jr. and R. J. Turyn, Research to develop the algebraic theory of codes, Report AFCRL-67-0365, Air Force Cambridge Res. Labs., Bedford, MA, 1967.

[3]

C. Bachoc and P. Gaborit, On extremal additive $GF(4)$-codes of lengths $10$ to $18$, J. Théorie Nombres Bordeaux, 12 (2000), 225-272. doi: 10.5802/jtnb.278.

[4]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes, 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.

[5]

J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des., 8 (2000), 174-188. doi: 10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T.

[6]

G. Birkhoff and S. MacLane, "A Survey of Modern Algebra,'' $4^{th}$ edition, MacMillan Publishing, New York, 1977.

[7]

A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387. doi: 10.1109/18.681315.

[8]

L. E. Danielsen, Graph-based classification of self-dual additive codes over finite fields, Adv. Math. Commun., 3 (2009), 329-348. doi: 10.3934/amc.2009.3.329.

[9]

L. E. Danielsen, On the classification of Hermitian self-dual additive codes over $GF(9)$, IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511. doi: 10.1109/TIT.2012.2196255.

[10]

L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over $GF(4)$ of length up to $12$, J. Comb. Theory Ser. A, 113 (2006), 1351-1367. doi: 10.1016/j.jcta.2005.12.004.

[11]

B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach, Des. Codes Cryptogr., 34 (2005), 89-116. doi: 10.1007/s10623-003-4196-x.

[12]

L. Dornhoff, "Group Representation Theory (Part A),'' Marcel Dekker, New York, 1971.

[13]

C. Drees, M. Epkenhans and M. Krüskemper, On the computation of the trace form of some Galois extensions, J. Algebra, 192 (1997), 209-234. doi: 10.1006/jabr.1996.6939.

[14]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes, Des. Codes Cryptogr., 18 (1999), 125-148. doi: 10.1023/A:1008389220478.

[15]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths $20$ and $22$, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.

[16]

P. Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, On additive $GF(4)$ codes, in "Codes and Association Schemes: DIMACS Workshop'' (eds. A. Barg and S. Litsyn), Amer. Math. Soc., Providence, (2001), 135-149.

[17]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrés Internl. de Mathématique, Gauthier-Villars, Paris, (1971), 211-215.

[18]

G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y.

[19]

S. K. Houghten, C. W. H. Lam and L. H. Thiel, Construction of $(48,24,12)$ doubly-even self-dual codes, Congr. Numer., 103 (1994), 41-53.

[20]

S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extended quadratic residue code is the only $(48,24,12)$ self-dual doubly-even code, IEEE Trans. Inform. Theory, IT-49 (2003), 53-59. doi: 10.1109/TIT.2002.806146.

[21]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$, Adv. Math. Commun., 1 (2007), 429-461. doi: 10.3934/amc.2007.1.427.

[22]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$ of even length, Adv. Math. Commun., 2 (2008), 309-343. doi: 10.3934/amc.2008.2.309.

[23]

W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284. doi: 10.1504/IJICOT.2010.032543.

[24]

W. C. Huffman, Self-dual $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes with an automorphism of prime order, Adv. Math. Commun., 7 (2013), 57-90. doi: 10.3934/amc.2013.7.57.

[25]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.

[26]

J.-L. Kim and X. Liu, A generalized Gleason-Pierce-Ward theorem, Des. Codes Cryptogr., 52 (2009), 363-380. doi: 10.1007/s10623-009-9286-y.

[27]

J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves, Discrete Math., 308 (2008), 3115-3124. doi: 10.1016/j.disc.2007.08.038.

[28]

H. Koch, Unimodular lattices and self-dual codes, in "Proc. Intern. Congress Math.,'' Amer. Math. Soc., Providence, (1987), 457-465.

[29]

T. Y. Lam, "The Algebraic Theory of Quadratic Forms,'' Reading MA: WA Benjamin, 1973.

[30]

F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J., 42 (1963), 79-94.

[31]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' Elsevier, New York, 1977.

[32]

V. Pless, Power moment identities on weight distributions in error correcting codes, Inform. Control, 6 (1963), 147-152. doi: 10.1016/S0019-9958(63)90189-X.

[33]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832. doi: 10.1109/18.782103.

[34]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.

[35]

J. J. Rotman, "An Introduction to the Theory of Groups,'' $4^{th}$ edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.

[36]

N. J. A. Sloane, Relations between combinatorics and other parts of mathematics, Proc. Sympos. Pure Math., 34 (1979), 273-308.

[37]

D. E. Taylor, "The Geometry of the Classical Groups,'' Heldermann Verlag, Berlin, 1992.

[38]

H. N. Ward, Divisible codes, Arch. Math. (Basel), 36 (1981), 485-494. doi: 10.1007/BF01223730.

[39]

H. N. Ward, Quadratic residue codes and divisibility, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 827-870.

show all references

References:
[1]

I. M. Araújo, et al., GAP Reference Manual, The GAP Group, http://www.gap-system.org

[2]

E. F. Assmus, Jr., H. F. Mattson, Jr. and R. J. Turyn, Research to develop the algebraic theory of codes, Report AFCRL-67-0365, Air Force Cambridge Res. Labs., Bedford, MA, 1967.

[3]

C. Bachoc and P. Gaborit, On extremal additive $GF(4)$-codes of lengths $10$ to $18$, J. Théorie Nombres Bordeaux, 12 (2000), 225-272. doi: 10.5802/jtnb.278.

[4]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes, 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.

[5]

J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des., 8 (2000), 174-188. doi: 10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T.

[6]

G. Birkhoff and S. MacLane, "A Survey of Modern Algebra,'' $4^{th}$ edition, MacMillan Publishing, New York, 1977.

[7]

A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387. doi: 10.1109/18.681315.

[8]

L. E. Danielsen, Graph-based classification of self-dual additive codes over finite fields, Adv. Math. Commun., 3 (2009), 329-348. doi: 10.3934/amc.2009.3.329.

[9]

L. E. Danielsen, On the classification of Hermitian self-dual additive codes over $GF(9)$, IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511. doi: 10.1109/TIT.2012.2196255.

[10]

L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over $GF(4)$ of length up to $12$, J. Comb. Theory Ser. A, 113 (2006), 1351-1367. doi: 10.1016/j.jcta.2005.12.004.

[11]

B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach, Des. Codes Cryptogr., 34 (2005), 89-116. doi: 10.1007/s10623-003-4196-x.

[12]

L. Dornhoff, "Group Representation Theory (Part A),'' Marcel Dekker, New York, 1971.

[13]

C. Drees, M. Epkenhans and M. Krüskemper, On the computation of the trace form of some Galois extensions, J. Algebra, 192 (1997), 209-234. doi: 10.1006/jabr.1996.6939.

[14]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes, Des. Codes Cryptogr., 18 (1999), 125-148. doi: 10.1023/A:1008389220478.

[15]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths $20$ and $22$, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.

[16]

P. Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, On additive $GF(4)$ codes, in "Codes and Association Schemes: DIMACS Workshop'' (eds. A. Barg and S. Litsyn), Amer. Math. Soc., Providence, (2001), 135-149.

[17]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrés Internl. de Mathématique, Gauthier-Villars, Paris, (1971), 211-215.

[18]

G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y.

[19]

S. K. Houghten, C. W. H. Lam and L. H. Thiel, Construction of $(48,24,12)$ doubly-even self-dual codes, Congr. Numer., 103 (1994), 41-53.

[20]

S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extended quadratic residue code is the only $(48,24,12)$ self-dual doubly-even code, IEEE Trans. Inform. Theory, IT-49 (2003), 53-59. doi: 10.1109/TIT.2002.806146.

[21]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$, Adv. Math. Commun., 1 (2007), 429-461. doi: 10.3934/amc.2007.1.427.

[22]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$ of even length, Adv. Math. Commun., 2 (2008), 309-343. doi: 10.3934/amc.2008.2.309.

[23]

W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284. doi: 10.1504/IJICOT.2010.032543.

[24]

W. C. Huffman, Self-dual $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes with an automorphism of prime order, Adv. Math. Commun., 7 (2013), 57-90. doi: 10.3934/amc.2013.7.57.

[25]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.

[26]

J.-L. Kim and X. Liu, A generalized Gleason-Pierce-Ward theorem, Des. Codes Cryptogr., 52 (2009), 363-380. doi: 10.1007/s10623-009-9286-y.

[27]

J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves, Discrete Math., 308 (2008), 3115-3124. doi: 10.1016/j.disc.2007.08.038.

[28]

H. Koch, Unimodular lattices and self-dual codes, in "Proc. Intern. Congress Math.,'' Amer. Math. Soc., Providence, (1987), 457-465.

[29]

T. Y. Lam, "The Algebraic Theory of Quadratic Forms,'' Reading MA: WA Benjamin, 1973.

[30]

F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J., 42 (1963), 79-94.

[31]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' Elsevier, New York, 1977.

[32]

V. Pless, Power moment identities on weight distributions in error correcting codes, Inform. Control, 6 (1963), 147-152. doi: 10.1016/S0019-9958(63)90189-X.

[33]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832. doi: 10.1109/18.782103.

[34]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.

[35]

J. J. Rotman, "An Introduction to the Theory of Groups,'' $4^{th}$ edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.

[36]

N. J. A. Sloane, Relations between combinatorics and other parts of mathematics, Proc. Sympos. Pure Math., 34 (1979), 273-308.

[37]

D. E. Taylor, "The Geometry of the Classical Groups,'' Heldermann Verlag, Berlin, 1992.

[38]

H. N. Ward, Divisible codes, Arch. Math. (Basel), 36 (1981), 485-494. doi: 10.1007/BF01223730.

[39]

H. N. Ward, Quadratic residue codes and divisibility, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 827-870.

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