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May  2016, 10(2): 429-436. doi: 10.3934/amc.2016016

Coherence of sensing matrices coming from algebraic-geometric codes

Seungkook Park 1,

1. 

Department of Mathematics, Sookmyung Women's University, Cheongpa-ro 47 gil 100, Yongsan-Ku, Seoul 140-742, South Korea

Received  September 2014 Published  April 2016

Compressed sensing is a technique which is to used to reconstruct a sparse signal given few measurements of the signal. One of the main problems in compressed sensing is the deterministic construction of the sensing matrix. Li et al. introduced a new deterministic construction via algebraic-geometric codes (AG codes) and gave an upper bound for the coherence of the sensing matrices coming from AG codes. In this paper, we give the exact value of the coherence of the sensing matrices coming from AG codes in terms of the minimum distance of AG codes and deduce the upper bound given by Li et al. We also give formulas for the coherence of the sensing matrices coming from Hermitian two-point codes.
Keywords: algebraic-geometric code, minimum distance., coherence, Compressed sensing.
Mathematics Subject Classification: Primary: 11T71, 14G50; Secondary: 92C55, 11R58, 14H5.
Citation: Seungkook Park. Coherence of sensing matrices coming from algebraic-geometric codes. Advances in Mathematics of Communications, 2016, 10 (2) : 429-436. doi: 10.3934/amc.2016016
References:
[1]

P. Beelen, The order bound for general algebraic geometric codes, Finite Fields Appl., 13 (2007), 665-680. doi: 10.1016/j.ffa.2006.09.006.  Google Scholar

[2]

J. Bourgain, S. Dilworth, K. Ford, S. Konyagin and D. Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. J., 159 (2011), 145-185. doi: 10.1215/00127094-1384809.  Google Scholar

[3]

E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.  Google Scholar

[4]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.  Google Scholar

[5]

R. A. DeVore, Deterministic constructions of compressed sensing matrices, J. Complexity, 23 (2007), 918-925. doi: 10.1016/j.jco.2007.04.002.  Google Scholar

[6]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[7]

M. Homma and S. J. Kim, Toward the determination of the minimum distance of two-point codes on a Hermitian curve, Des. Codes Crypt., 37 (2005), 111-132. doi: 10.1007/s10623-004-3807-5.  Google Scholar

[8]

M. Homma and S. J. Kim, The complete determination of the minimum distance of two-point codes on a Hermitian curve, Des. Codes Crypt., 40 (2006), 5-24. doi: 10.1007/s10623-005-4599-y.  Google Scholar

[9]

M. Homma and S. J. Kim, The two-point codes on a Hermitian curve with the designed minimum distance, Des. Codes Crypt., 38 (2006), 55-81. doi: 10.1007/s10623-004-5661-x.  Google Scholar

[10]

M. Homma and S. J. Kim, The two-point codes with the designed distance on a Hermitian curve in even characteristic, Des. Codes Crypt., 39 (2006), 375-386. doi: 10.1007/s10623-005-5471-9.  Google Scholar

[11]

C. Kirfel and R. Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inf. Theory, 41 (1995), 1720-1732. doi: 10.1109/18.476245.  Google Scholar

[12]

S. Li, F. Gao, G. Ge and S. Zhang, Deterministic construction of compressed sensing matrices via algebraic curves, IEEE Trans. Inf. Theory, 58 (2012), 5035-5041. doi: 10.1109/TIT.2012.2196256.  Google Scholar

[13]

G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Des. Codes Crypt., 22 (2001), 107-121. doi: 10.1023/A:1008311518095.  Google Scholar

[14]

S. Park, Minimum distance of Hermitian two-point codes, Des. Codes Crypt., 57 (2010), 195-213. doi: 10.1007/s10623-009-9361-4.  Google Scholar

[15]

H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, Berlin, 2009.  Google Scholar

[16]

C. P. Xing and H. Chen, Improvements on parameters of one-point AG codes from Hermitian curves, IEEE Trans. Inf. Theory, 48 (2002), 535-537. doi: 10.1109/18.979330.  Google Scholar

show all references

References:
[1]

P. Beelen, The order bound for general algebraic geometric codes, Finite Fields Appl., 13 (2007), 665-680. doi: 10.1016/j.ffa.2006.09.006.  Google Scholar

[2]

J. Bourgain, S. Dilworth, K. Ford, S. Konyagin and D. Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. J., 159 (2011), 145-185. doi: 10.1215/00127094-1384809.  Google Scholar

[3]

E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.  Google Scholar

[4]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.  Google Scholar

[5]

R. A. DeVore, Deterministic constructions of compressed sensing matrices, J. Complexity, 23 (2007), 918-925. doi: 10.1016/j.jco.2007.04.002.  Google Scholar

[6]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[7]

M. Homma and S. J. Kim, Toward the determination of the minimum distance of two-point codes on a Hermitian curve, Des. Codes Crypt., 37 (2005), 111-132. doi: 10.1007/s10623-004-3807-5.  Google Scholar

[8]

M. Homma and S. J. Kim, The complete determination of the minimum distance of two-point codes on a Hermitian curve, Des. Codes Crypt., 40 (2006), 5-24. doi: 10.1007/s10623-005-4599-y.  Google Scholar

[9]

M. Homma and S. J. Kim, The two-point codes on a Hermitian curve with the designed minimum distance, Des. Codes Crypt., 38 (2006), 55-81. doi: 10.1007/s10623-004-5661-x.  Google Scholar

[10]

M. Homma and S. J. Kim, The two-point codes with the designed distance on a Hermitian curve in even characteristic, Des. Codes Crypt., 39 (2006), 375-386. doi: 10.1007/s10623-005-5471-9.  Google Scholar

[11]

C. Kirfel and R. Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inf. Theory, 41 (1995), 1720-1732. doi: 10.1109/18.476245.  Google Scholar

[12]

S. Li, F. Gao, G. Ge and S. Zhang, Deterministic construction of compressed sensing matrices via algebraic curves, IEEE Trans. Inf. Theory, 58 (2012), 5035-5041. doi: 10.1109/TIT.2012.2196256.  Google Scholar

[13]

G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Des. Codes Crypt., 22 (2001), 107-121. doi: 10.1023/A:1008311518095.  Google Scholar

[14]

S. Park, Minimum distance of Hermitian two-point codes, Des. Codes Crypt., 57 (2010), 195-213. doi: 10.1007/s10623-009-9361-4.  Google Scholar

[15]

H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, Berlin, 2009.  Google Scholar

[16]

C. P. Xing and H. Chen, Improvements on parameters of one-point AG codes from Hermitian curves, IEEE Trans. Inf. Theory, 48 (2002), 535-537. doi: 10.1109/18.979330.  Google Scholar

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