Self-dual \mathbb{F}_q-linear \mathbb{F}_{q^t}-codes with an automorphism of prime order

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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

Abstract
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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, http://www.gap-system.org
[2]	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.
[3]	J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des., 8 (2000), 174-188.
[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), 163-169. doi: 10.1023/A:1022588407585.
[5]	S. Bouyuklieva, A. Malevich and W. Willems, Automorphisms of extremal self-dual codes, IEEE Trans. Inform. Theory, IT-56 (2010), 2091-2096.
[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), 605-615. doi: 10.1016/j.ffa.2006.01.001.
[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), 21-37.
[8]	S. Buyuklieva and V. Yorgov, Singly-even self-dual codes of length 40, Des. Codes Crypt., 9 (1996), 131-141.
[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), 1369-1387.
[10]	L. E. Danielsen, Graph-based classification of self-dual additive codes over finite fields, Adv. Math. Commun., 3 (2009), 329-348.
[11]	L. E. Danielsen, On the classification of Hermitian self-dual additive codes over GF(9), IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511.
[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), 89-116.
[13]	R. Dontcheva and M. Harada, Extremal doubly-even $[80,40,16]$ codes with an automorphism of order 19, Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/S1071-5797(02)00018-7.
[14]	R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.
[15]	W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.
[16]	W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 I, IEEE Trans. Inform. Theory, IT-36 (1990), 651-660.
[17]	W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 II, IEEE Trans. Inform. Theory, IT-37 (1991), 1206-1216.
[18]	W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40, IEEE Trans. Inform. Theory, IT-38 (1992), 1395-1400.
[19]	W. C. Huffman, Decompositions and extremal type II codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, IT-44 (1998), 800-809.
[20]	W. C. Huffman, Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order, Adv. Math. Commun., 1 (2007), 357-398.
[21]	W. C. Huffman, Additive cyclic codes over $\mathbb F_4$, Adv. Math. Commun., 1 (2007), 429-461.
[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), 681-712. doi: 10.1016/j.ffa.2006.02.003.
[23]	W. C. Huffman, Additive cyclic codes over $\mathbb F_4$ of even length, Adv. Math. Commun., 2 (2008), 309-343.
[24]	W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inform. Coding Theory, 1 (2010), 249-284.
[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), 97-106. doi: 10.1007/BF01398008.
[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), 341-349. doi: 10.1006/ffta.2000.0295.
[27]	V. I. Iorgov, Binary self-dual codes with automorphisms of odd order, Problems Inform. Trans., 19 (1983), 260-270.
[28]	V. I. Iorgov, Doubly even extremal codes of length 64, Problems Inform. Trans., 22 (1986), 277-284.
[29]	J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves, Discrete Math., 308 (2008), 3115-3124.
[30]	E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832.
[31]	R. P. Russeva, Self-dual $[24,12,8]$ quaternary codes with a nontrivial automorphism of order 3, Finite Fields Appl., 8 (2002), 34-51. doi: 10.1006/ffta.2001.0322.
[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), 7498-7506.
[33]	V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inform. Theory, IT-33 (1987), 77-82.
[34]	V. Y. Yorgov, The extremal codes of length 42 with an automorphism of order 7, Discrete Math., 190 (1998), 201-213.
[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), 253-257.

show all references

References:

[1]	I. M. Araújo, et al., GAP reference manual, The GAP Group, http://www.gap-system.org
[2]	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.
[3]	J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des., 8 (2000), 174-188.
[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), 163-169. doi: 10.1023/A:1022588407585.
[5]	S. Bouyuklieva, A. Malevich and W. Willems, Automorphisms of extremal self-dual codes, IEEE Trans. Inform. Theory, IT-56 (2010), 2091-2096.
[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), 605-615. doi: 10.1016/j.ffa.2006.01.001.
[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), 21-37.
[8]	S. Buyuklieva and V. Yorgov, Singly-even self-dual codes of length 40, Des. Codes Crypt., 9 (1996), 131-141.
[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), 1369-1387.
[10]	L. E. Danielsen, Graph-based classification of self-dual additive codes over finite fields, Adv. Math. Commun., 3 (2009), 329-348.
[11]	L. E. Danielsen, On the classification of Hermitian self-dual additive codes over GF(9), IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511.
[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), 89-116.
[13]	R. Dontcheva and M. Harada, Extremal doubly-even $[80,40,16]$ codes with an automorphism of order 19, Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/S1071-5797(02)00018-7.
[14]	R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.
[15]	W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.
[16]	W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 I, IEEE Trans. Inform. Theory, IT-36 (1990), 651-660.
[17]	W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28 II, IEEE Trans. Inform. Theory, IT-37 (1991), 1206-1216.
[18]	W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40, IEEE Trans. Inform. Theory, IT-38 (1992), 1395-1400.
[19]	W. C. Huffman, Decompositions and extremal type II codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, IT-44 (1998), 800-809.
[20]	W. C. Huffman, Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order, Adv. Math. Commun., 1 (2007), 357-398.
[21]	W. C. Huffman, Additive cyclic codes over $\mathbb F_4$, Adv. Math. Commun., 1 (2007), 429-461.
[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), 681-712. doi: 10.1016/j.ffa.2006.02.003.
[23]	W. C. Huffman, Additive cyclic codes over $\mathbb F_4$ of even length, Adv. Math. Commun., 2 (2008), 309-343.
[24]	W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inform. Coding Theory, 1 (2010), 249-284.
[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), 97-106. doi: 10.1007/BF01398008.
[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), 341-349. doi: 10.1006/ffta.2000.0295.
[27]	V. I. Iorgov, Binary self-dual codes with automorphisms of odd order, Problems Inform. Trans., 19 (1983), 260-270.
[28]	V. I. Iorgov, Doubly even extremal codes of length 64, Problems Inform. Trans., 22 (1986), 277-284.
[29]	J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves, Discrete Math., 308 (2008), 3115-3124.
[30]	E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832.
[31]	R. P. Russeva, Self-dual $[24,12,8]$ quaternary codes with a nontrivial automorphism of order 3, Finite Fields Appl., 8 (2002), 34-51. doi: 10.1006/ffta.2001.0322.
[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), 7498-7506.
[33]	V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inform. Theory, IT-33 (1987), 77-82.
[34]	V. Y. Yorgov, The extremal codes of length 42 with an automorphism of order 7, Discrete Math., 190 (1998), 201-213.
[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), 253-257.

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