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

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 $t=2$.
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]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes,, in, (2007), 276.   Google Scholar

[3]

J. Bierbrauer and Y. Edel, Quantum twisted codes,, J. Combin. Des., 8 (2000), 174.   Google Scholar

[4]

S. Bouyuklieva and M. Harada, Extremal self-dual $[50,25,10]$ codes with automorphisms of order 3 and quasi-symmetric 2-$(49,9,6)$ designs,, Des. Codes Crypt., 28 (2003), 163.  doi: 10.1023/A:1022588407585.  Google Scholar

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S. Bouyuklieva, A. Malevich and W. Willems, Automorphisms of extremal self-dual codes,, IEEE Trans. Inform. Theory, IT-56 (2010), 2091.   Google Scholar

[6]

S. Bouyuklieva, N. Yankov and R. Russeva, Classification of the binary self-dual $[42,21,8]$ codes having an automorphism of order 3,, Finite Fields Appl., 13 (2007), 605.  doi: 10.1016/j.ffa.2006.01.001.  Google Scholar

[7]

S. Bouyuklieva, N. Yankov and R. Russeva, On the classication of binary self-dual $[44,22,8]$ codes with an automorphism of order 3 or 7,, Int. J. Inform. Coding Theory, 2 (2011), 21.   Google Scholar

[8]

S. Buyuklieva and V. Yorgov, Singly-even self-dual codes of length 40,, Des. Codes Crypt., 9 (1996), 131.   Google Scholar

[9]

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.   Google Scholar

[10]

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

[11]

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

[12]

B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach,, Des. Codes Crypt., 34 (2005), 89.   Google Scholar

[13]

R. Dontcheva and M. Harada, Extremal doubly-even $[80,40,16]$ codes with an automorphism of order 19,, Finite Fields Appl., 9 (2003), 157.  doi: 10.1016/S1071-5797(02)00018-7.  Google Scholar

[14]

R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7,, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75.  doi: 10.1007/s00200-003-0126-4.  Google Scholar

[15]

W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48,, IEEE Trans. Inform. Theory, IT-28 (1982), 511.   Google Scholar

[16]

W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 I,, IEEE Trans. Inform. Theory, IT-36 (1990), 651.   Google Scholar

[17]

W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 II,, IEEE Trans. Inform. Theory, IT-37 (1991), 1206.   Google Scholar

[18]

W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40,, IEEE Trans. Inform. Theory, IT-38 (1992), 1395.   Google Scholar

[19]

W. C. Huffman, Decompositions and extremal type II codes over $\mathbb Z_4$,, IEEE Trans. Inform. Theory, IT-44 (1998), 800.   Google Scholar

[20]

W. C. Huffman, Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order,, Adv. Math. Commun., 1 (2007), 357.   Google Scholar

[21]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$,, Adv. Math. Commun., 1 (2007), 429.   Google Scholar

[22]

W. C. Huffman, On the decomposition of self-dual codes over $\mathbb F_2 + u\mathbb F_2$ with an automorphism of odd prime order,, Finite Fields Appl., 13 (2007), 681.  doi: 10.1016/j.ffa.2006.02.003.  Google Scholar

[23]

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

[24]

W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes,, Int. J. Inform. Coding Theory, 1 (2010), 249.   Google Scholar

[25]

W. C. Huffman and V. D. Tonchev, The existence of extremal self-dual $[50,25,10]$ codes and quasi-symmetric 2-$(49,9,6)$ designs,, Des. Codes Crypt., 6 (1995), 97.  doi: 10.1007/BF01398008.  Google Scholar

[26]

W. C. Huffman and V. D. Tonchev, The $[52,26,10]$ binary self-dual codes with an automorphism of order 7,, Finite Fields Appl., 7 (2001), 341.  doi: 10.1006/ffta.2000.0295.  Google Scholar

[27]

V. I. Iorgov, Binary self-dual codes with automorphisms of odd order,, Problems Inform. Trans., 19 (1983), 260.   Google Scholar

[28]

V. I. Iorgov, Doubly even extremal codes of length 64,, Problems Inform. Trans., 22 (1986), 277.   Google Scholar

[29]

J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves,, Discrete Math., 308 (2008), 3115.   Google Scholar

[30]

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

[31]

R. P. Russeva, Self-dual $[24,12,8]$ quaternary codes with a nontrivial automorphism of order 3,, Finite Fields Appl., 8 (2002), 34.  doi: 10.1006/ffta.2001.0322.  Google Scholar

[32]

N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13,, IEEE Trans. Inform. Theory, IT-57 (2011), 7498.   Google Scholar

[33]

V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56,, IEEE Trans. Inform. Theory, IT-33 (1987), 77.   Google Scholar

[34]

V. Y. Yorgov, The extremal codes of length 42 with an automorphism of order 7,, Discrete Math., 190 (1998), 201.   Google Scholar

[35]

V. Y. Yorgov and N. Ziapov, Doubly even self-dual $[40,20,8]$ codes with automorphism of an odd order,, Problems Inform. Trans., 32 (1996), 253.   Google Scholar

show all references

References:
[1]

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

[2]

J. Bierbrauer, Cyclic additive and quantum stabilizer codes,, in, (2007), 276.   Google Scholar

[3]

J. Bierbrauer and Y. Edel, Quantum twisted codes,, J. Combin. Des., 8 (2000), 174.   Google Scholar

[4]

S. Bouyuklieva and M. Harada, Extremal self-dual $[50,25,10]$ codes with automorphisms of order 3 and quasi-symmetric 2-$(49,9,6)$ designs,, Des. Codes Crypt., 28 (2003), 163.  doi: 10.1023/A:1022588407585.  Google Scholar

[5]

S. Bouyuklieva, A. Malevich and W. Willems, Automorphisms of extremal self-dual codes,, IEEE Trans. Inform. Theory, IT-56 (2010), 2091.   Google Scholar

[6]

S. Bouyuklieva, N. Yankov and R. Russeva, Classification of the binary self-dual $[42,21,8]$ codes having an automorphism of order 3,, Finite Fields Appl., 13 (2007), 605.  doi: 10.1016/j.ffa.2006.01.001.  Google Scholar

[7]

S. Bouyuklieva, N. Yankov and R. Russeva, On the classication of binary self-dual $[44,22,8]$ codes with an automorphism of order 3 or 7,, Int. J. Inform. Coding Theory, 2 (2011), 21.   Google Scholar

[8]

S. Buyuklieva and V. Yorgov, Singly-even self-dual codes of length 40,, Des. Codes Crypt., 9 (1996), 131.   Google Scholar

[9]

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.   Google Scholar

[10]

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

[11]

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

[12]

B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach,, Des. Codes Crypt., 34 (2005), 89.   Google Scholar

[13]

R. Dontcheva and M. Harada, Extremal doubly-even $[80,40,16]$ codes with an automorphism of order 19,, Finite Fields Appl., 9 (2003), 157.  doi: 10.1016/S1071-5797(02)00018-7.  Google Scholar

[14]

R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7,, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75.  doi: 10.1007/s00200-003-0126-4.  Google Scholar

[15]

W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48,, IEEE Trans. Inform. Theory, IT-28 (1982), 511.   Google Scholar

[16]

W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 I,, IEEE Trans. Inform. Theory, IT-36 (1990), 651.   Google Scholar

[17]

W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 II,, IEEE Trans. Inform. Theory, IT-37 (1991), 1206.   Google Scholar

[18]

W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40,, IEEE Trans. Inform. Theory, IT-38 (1992), 1395.   Google Scholar

[19]

W. C. Huffman, Decompositions and extremal type II codes over $\mathbb Z_4$,, IEEE Trans. Inform. Theory, IT-44 (1998), 800.   Google Scholar

[20]

W. C. Huffman, Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order,, Adv. Math. Commun., 1 (2007), 357.   Google Scholar

[21]

W. C. Huffman, Additive cyclic codes over $\mathbb F_4$,, Adv. Math. Commun., 1 (2007), 429.   Google Scholar

[22]

W. C. Huffman, On the decomposition of self-dual codes over $\mathbb F_2 + u\mathbb F_2$ with an automorphism of odd prime order,, Finite Fields Appl., 13 (2007), 681.  doi: 10.1016/j.ffa.2006.02.003.  Google Scholar

[23]

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

[24]

W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes,, Int. J. Inform. Coding Theory, 1 (2010), 249.   Google Scholar

[25]

W. C. Huffman and V. D. Tonchev, The existence of extremal self-dual $[50,25,10]$ codes and quasi-symmetric 2-$(49,9,6)$ designs,, Des. Codes Crypt., 6 (1995), 97.  doi: 10.1007/BF01398008.  Google Scholar

[26]

W. C. Huffman and V. D. Tonchev, The $[52,26,10]$ binary self-dual codes with an automorphism of order 7,, Finite Fields Appl., 7 (2001), 341.  doi: 10.1006/ffta.2000.0295.  Google Scholar

[27]

V. I. Iorgov, Binary self-dual codes with automorphisms of odd order,, Problems Inform. Trans., 19 (1983), 260.   Google Scholar

[28]

V. I. Iorgov, Doubly even extremal codes of length 64,, Problems Inform. Trans., 22 (1986), 277.   Google Scholar

[29]

J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves,, Discrete Math., 308 (2008), 3115.   Google Scholar

[30]

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

[31]

R. P. Russeva, Self-dual $[24,12,8]$ quaternary codes with a nontrivial automorphism of order 3,, Finite Fields Appl., 8 (2002), 34.  doi: 10.1006/ffta.2001.0322.  Google Scholar

[32]

N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13,, IEEE Trans. Inform. Theory, IT-57 (2011), 7498.   Google Scholar

[33]

V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56,, IEEE Trans. Inform. Theory, IT-33 (1987), 77.   Google Scholar

[34]

V. Y. Yorgov, The extremal codes of length 42 with an automorphism of order 7,, Discrete Math., 190 (1998), 201.   Google Scholar

[35]

V. Y. Yorgov and N. Ziapov, Doubly even self-dual $[40,20,8]$ codes with automorphism of an odd order,, Problems Inform. Trans., 32 (1996), 253.   Google Scholar

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