# American Institute of Mathematical Sciences

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:
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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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##### 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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