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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

W. Cary Huffman 1,

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.
Keywords: self-dual codes, MacWilliams identities, MDS codes, Additive codes, mass formulas..
Mathematics Subject Classification: Primary: 94B60, 94B05; Secondary: 94B2.
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]

E. F. Assmus, Jr., H. F. Mattson, Jr. and R. J. Turyn, Research to develop the algebraic theory of codes,, Report AFCRL-67-0365, (1967), 67. Google Scholar

[3]

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

[4]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes,, in, (2007), 276. doi: 10.1007/978-3-540-73074-3_21. Google Scholar

[5]

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

[6]

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

[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. doi: 10.1109/18.681315. Google Scholar

[8]

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

[9]

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

[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. doi: 10.1016/j.jcta.2005.12.004. Google Scholar

[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. doi: 10.1007/s10623-003-4196-x. Google Scholar

[12]

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

[13]

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

[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. doi: 10.1023/A:1008389220478. Google Scholar

[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. doi: 10.1016/S0166-218X(00)00345-0. Google Scholar

[16]

P. Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, On additive $GF(4)$ codes,, in, (2001), 135. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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. doi: 10.1109/TIT.2002.806146. Google Scholar

[21]

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

[22]

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

[23]

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

[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. doi: 10.3934/amc.2013.7.57. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

H. Koch, Unimodular lattices and self-dual codes,, in, (1987), 457. Google Scholar

[29]

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

[30]

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

[31]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'', Elsevier, (1977). Google Scholar

[32]

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

[33]

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

[34]

E. M. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177. Google Scholar

[35]

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

[36]

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

[37]

D. E. Taylor, "The Geometry of the Classical Groups,'', Heldermann Verlag, (1992). Google Scholar

[38]

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

[39]

H. N. Ward, Quadratic residue codes and divisibility,, in, (1998), 827. Google Scholar

show all references

References:
[1]

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

[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, (1967), 67. Google Scholar

[3]

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

[4]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes,, in, (2007), 276. doi: 10.1007/978-3-540-73074-3_21. Google Scholar

[5]

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

[6]

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

[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. doi: 10.1109/18.681315. Google Scholar

[8]

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

[9]

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

[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. doi: 10.1016/j.jcta.2005.12.004. Google Scholar

[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. doi: 10.1007/s10623-003-4196-x. Google Scholar

[12]

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

[13]

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

[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. doi: 10.1023/A:1008389220478. Google Scholar

[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. doi: 10.1016/S0166-218X(00)00345-0. Google Scholar

[16]

P. Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, On additive $GF(4)$ codes,, in, (2001), 135. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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. doi: 10.1109/TIT.2002.806146. Google Scholar

[21]

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

[22]

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

[23]

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

[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. doi: 10.3934/amc.2013.7.57. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

H. Koch, Unimodular lattices and self-dual codes,, in, (1987), 457. Google Scholar

[29]

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

[30]

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

[31]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'', Elsevier, (1977). Google Scholar

[32]

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

[33]

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

[34]

E. M. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177. Google Scholar

[35]

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

[36]

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

[37]

D. E. Taylor, "The Geometry of the Classical Groups,'', Heldermann Verlag, (1992). Google Scholar

[38]

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

[39]

H. N. Ward, Quadratic residue codes and divisibility,, in, (1998), 827. Google Scholar

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