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New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings
Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes
1. | College of Applied Sciences, Beijing University of Technology, Beijing 100124, China |
2. | Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China |
Let $ \mathbb{F}_{q^t} $ be a finite field of cardinality $ q^t $, where $ q $ is a power of a prime number $ p $ and $ t\geq 1 $ is a positive integer. Firstly, a family of cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes of length $ n $ is given, where $ n $ is a positive integer coprime to $ q $. Then according to the structure of this kind of codes, we construct $ 60 $ optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^2} $-codes which have the same parameters as the MDS codes over $ \mathbb{F}_{q^2} $.
References:
[1] |
T. L. Alderson,
Extending MDS codes, Ann. Comb., 9 (2005), 125-135.
doi: 10.1007/s00026-005-0245-7. |
[2] |
I. Bouyukliev and J. Simonis,
Some new results on optimal codes over $\mathbb{F}_5$, Des. Codes Cryptogr., 30 (2003), 97-111.
doi: 10.1023/A:1024763410967. |
[3] |
S. Bouyuklieva and P. R. J. Östergảrd,
New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.
doi: 10.1007/s10623-006-0018-2. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, J. Symb. Comput., 24 (1997), 235-265. Google Scholar |
[5] |
Y. L. Cao and Y. Gao,
Repeated root cyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Finite Fields Appl., 31 (2015), 202-227.
doi: 10.1016/j.ffa.2014.10.003. |
[6] |
Y. L. Cao, X. X. Chang and Y. Cao,
Constacyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Appl. Algebra Engrg. Comm. Comput., 26 (2015), 369-388.
doi: 10.1007/s00200-015-0257-4. |
[7] |
Y. L. Cao, J. Gao and F.-W. Fu,
Semisimple multivariable $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Des. Codes Cryptogr., 77 (2015), 153-177.
doi: 10.1007/s10623-014-9994-9. |
[8] |
B. C. Chen and H. W. Liu,
New constructions of MDS codes with complementary duals, IEEE Trans. Inform. Theory, 64 (2018), 5776-5782.
doi: 10.1109/TIT.2017.2748955. |
[9] |
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. |
[10] |
S. Dodunekov and I. Landgev,
On near-MDS codes, J. Geom., 54 (1995), 30-43.
doi: 10.1007/BF01222850. |
[11] |
R. Gabrys, E. Yaakobi, M. Blaum and P. H. Siegel,
Constructions of partial MDS codes over small fields, IEEE Internat. Symposium Inform. Theory, 65 (2019), 3692-3701.
doi: 10.1109/TIT.2018.2890201. |
[12] |
M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Internat. Symposium Inform. Theory, (2008), 1954-1957. Google Scholar |
[13] |
W. C. Huffman,
Cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Int. J. Inf. and Coding Theory, 1 (2010), 249-284.
doi: 10.1504/IJICOT.2010.032543. |
[14] |
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. |
[15] |
W. C. Huffman,
On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.
doi: 10.3934/amc.2013.7.349. |
[16] |
B. Hurley and T. Hurley,
Systems of MDS codes from units and idempotents, Discrete Math., 335 (2014), 81-91.
doi: 10.1016/j.disc.2014.07.010. |
[17] |
L. F. Jin, S. Ling, J. Q. Luo and C. P. Xing,
Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes, IEEE Trans. Inform. Theory, 56 (2010), 4735-4740.
doi: 10.1109/TIT.2010.2054174. |
[18] |
L. F. Jin and C. P. Xing,
New MDS self-dual codes from generalized Reed-Solomon codes, IEEE Trans. Inform. Theory, 63 (2017), 1434-1438.
doi: 10.1109/TIT.2016.2645759. |
[19] |
T. Maruta,
On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.
doi: 10.1006/S0195-6698(97)90000-7. |
[20] |
R. M. Roth and G. Seroussi,
On cyclic MDS codes of length $q$ over $GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.
doi: 10.1109/TIT.1986.1057151. |
[21] |
R. M. Roth and G. Seroussi,
On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.
doi: 10.1109/TIT.1985.1057113. |
[22] |
M. J. Shi and P. Solé,
Optimal $p$-ary codes from one-weight and two-weight codes over $\mathbb{F}_p+v{\mathbb{F}_p}^*$, J. Syst. Sci. Complex., 28 (2015), 679-690.
doi: 10.1007/s11424-015-3265-3. |
[23] |
Z.-X. Wan,
Cyclic codes over Galois rings$^*$, Algebra Colloquium, 6 (1999), 291-304.
|
show all references
References:
[1] |
T. L. Alderson,
Extending MDS codes, Ann. Comb., 9 (2005), 125-135.
doi: 10.1007/s00026-005-0245-7. |
[2] |
I. Bouyukliev and J. Simonis,
Some new results on optimal codes over $\mathbb{F}_5$, Des. Codes Cryptogr., 30 (2003), 97-111.
doi: 10.1023/A:1024763410967. |
[3] |
S. Bouyuklieva and P. R. J. Östergảrd,
New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.
doi: 10.1007/s10623-006-0018-2. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, J. Symb. Comput., 24 (1997), 235-265. Google Scholar |
[5] |
Y. L. Cao and Y. Gao,
Repeated root cyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Finite Fields Appl., 31 (2015), 202-227.
doi: 10.1016/j.ffa.2014.10.003. |
[6] |
Y. L. Cao, X. X. Chang and Y. Cao,
Constacyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Appl. Algebra Engrg. Comm. Comput., 26 (2015), 369-388.
doi: 10.1007/s00200-015-0257-4. |
[7] |
Y. L. Cao, J. Gao and F.-W. Fu,
Semisimple multivariable $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Des. Codes Cryptogr., 77 (2015), 153-177.
doi: 10.1007/s10623-014-9994-9. |
[8] |
B. C. Chen and H. W. Liu,
New constructions of MDS codes with complementary duals, IEEE Trans. Inform. Theory, 64 (2018), 5776-5782.
doi: 10.1109/TIT.2017.2748955. |
[9] |
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. |
[10] |
S. Dodunekov and I. Landgev,
On near-MDS codes, J. Geom., 54 (1995), 30-43.
doi: 10.1007/BF01222850. |
[11] |
R. Gabrys, E. Yaakobi, M. Blaum and P. H. Siegel,
Constructions of partial MDS codes over small fields, IEEE Internat. Symposium Inform. Theory, 65 (2019), 3692-3701.
doi: 10.1109/TIT.2018.2890201. |
[12] |
M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Internat. Symposium Inform. Theory, (2008), 1954-1957. Google Scholar |
[13] |
W. C. Huffman,
Cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Int. J. Inf. and Coding Theory, 1 (2010), 249-284.
doi: 10.1504/IJICOT.2010.032543. |
[14] |
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. |
[15] |
W. C. Huffman,
On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.
doi: 10.3934/amc.2013.7.349. |
[16] |
B. Hurley and T. Hurley,
Systems of MDS codes from units and idempotents, Discrete Math., 335 (2014), 81-91.
doi: 10.1016/j.disc.2014.07.010. |
[17] |
L. F. Jin, S. Ling, J. Q. Luo and C. P. Xing,
Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes, IEEE Trans. Inform. Theory, 56 (2010), 4735-4740.
doi: 10.1109/TIT.2010.2054174. |
[18] |
L. F. Jin and C. P. Xing,
New MDS self-dual codes from generalized Reed-Solomon codes, IEEE Trans. Inform. Theory, 63 (2017), 1434-1438.
doi: 10.1109/TIT.2016.2645759. |
[19] |
T. Maruta,
On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.
doi: 10.1006/S0195-6698(97)90000-7. |
[20] |
R. M. Roth and G. Seroussi,
On cyclic MDS codes of length $q$ over $GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.
doi: 10.1109/TIT.1986.1057151. |
[21] |
R. M. Roth and G. Seroussi,
On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.
doi: 10.1109/TIT.1985.1057113. |
[22] |
M. J. Shi and P. Solé,
Optimal $p$-ary codes from one-weight and two-weight codes over $\mathbb{F}_p+v{\mathbb{F}_p}^*$, J. Syst. Sci. Complex., 28 (2015), 679-690.
doi: 10.1007/s11424-015-3265-3. |
[23] |
Z.-X. Wan,
Cyclic codes over Galois rings$^*$, Algebra Colloquium, 6 (1999), 291-304.
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