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## An algorithm for list decoding number field codes

K

K

NTRODUCTION

1

n

1

k

1

2

n

1

n

1

n

K

1

n

1

n

K

K

K

1

K

n

1

n

i

i

i

i

q

i

i

i

i

K

i

i≤n

1

2

n

i≤n

i

i≤k

i

1/d

K

K

### This algorithm is polynomial in d , log(N), 1/ε and log | ∆ | . II. G

ENERALITIES ON NUMBER FIELDS

1

i

i≤r1

2

i

r1<i≤r1+2r2

2

K

K

R

r1

r2

(3)

i

R

2

R

2

i

i

i

2

2

i

i≤d

K

i

i

2

2

i

j

i

i

K

K

K

K

K

1

1

l

l

1

l

1

l

K

−1

K

−1

K

K

K

K

K

l

K

i

i

i≤n

i

i

1

1

n

n

i

i≤n

i

i

i≤n

ECODING WITH

OPPERSMITH

S THEOREM

K

K

1

d

K

K

i

i

i

K

β

i

i

i

−d2/2

β2/l

P

i≤klogN(pi) P

i≤nlogN(pi)

i≤n

i

i

i≤k

i

1/d

1

n

Kn

K

i

i

K

i≤k

i

n

1

loglogN(pN(pn1))

OHNSON

### -

TYPE BOUND FOR NUMBER FIELDS CODES

n

i=1

i

K

K

i

K

i

t

i=1

i

2Bd d

t

i=1

i

n

i=1

i

i

i

i:x6=y modpi

i

n

i=1

i

i

### V. G

ENERAL DESCRIPTION OF THE ALGORITHM

K

(4)

i≤k

i

1/d

i≤n

i

K

K

1

K

n

1

n

1

n

i

1

n

i

K

i

i

i

i

i

i

i

i

i≤n

K

i≤n

izi

K

i

i

K

i

i

ziiai

i

i

i

i

i

ziai

1/d

i≤n

i

i

i

i

i

ziai

1/d

K

1

n

1

n

i

K

i

i

i

i

1:

2:

i≤n

izi

K

3:

i

i

i

### VI. E

XISTENCE OF THE DECODING POLYNOMIAL

i

i≤n

i

izi

K

K

K

d/2

′d

r1+r2−1+γ

R

r1+2r2

i≤r1

i

i

r1<i≤r1+2r2

i

i

r1+2r2

R

r1+2r2

K

R

d

r2

d/2

′d

d,∆,γ

d/2

r1+r2−1+γ

K

d,∆,γ

l/2

d

l+1

0

1

l

l

i

i

i

i

i

d,∆,γ

i

d

i

d,∆,γ

l+1

l+1 l

i=0

−i

d

K

i

izi

i

izi

l/2

d,∆,γ

1/d

i

i

zi2+1

d(l+1)1

d,∆,γ

l/2

d

l+1

K

i

i

zi2+1

K

i

izi

d,∆,γ

l/2

d

l+1

K

i

izi

(5)

1

2

K

1

2

i

izi

1

2

1

2

### VII. C

OMPUTATION OF THE DECODING POLYNOMIAL

i

izi

K

i

izi

l+1

K

K

l+1

i

i

i

i

izi

i

izi

K

izi

i

izi

i

i

i≤n

i

izi

1

n

i

K

i

i

izi

dl2

2+d(6+3d)

3

2+112d

1:

2:

i

i

3:

i

ij

zii−j

ij

i

j

4:

i

ij

K

ij

j

i

zi

5:

ij

ij

j≤l+1

izi

6:

7:

i

i

i≤l+1

1

i

izi

8:

i

i

i≤l+1

2

0

1

l

1

0

1

l

l

2

9:

i

i

i≤l+1

2

10:

1

2

1

11:

1

1

1

2

### VIII. G

OOD WEIGHT SETTINGS

i

d,∆,γ

3−d2

3(1+d(2+d))

2+112d

d,∆,γ

1 d

1

n

i≤n

K

i

i≤k

i

1/d

i

K

i≤n

i

i

i

d2

d

i≤n

i

i

d,∆,γ

i

i

i

i

i

i

i

i≤k

i

1/d

1

n

i

K

i

K

in

i

i

i

d2

d

in

i2

i

max2

i

i

i

max

i

i

i

i

i

i≤n

i

i

i

d2

d

i≤n

2i

i

2

i

d,∆,γ

i

i2

A3

i

A22

i≤n

i

i

d2

d

d2

d

i≤n

i

i

i

i≤n

i

i

d2

d

d2

d

i≤n

i

i

d,∆,γ

(6)

10 logε N

loglogβd,∆,γN

d2

d

i≤n

i2

i

d2

d

i≤n

i2

i

1ε

1

n

i

i+1

k+1

k

2

log(2d

2Bd) logN(pk+1)

k+1

k

2

i

i

i

k+1

k+1

d2

d

k+1

k+1

n

i=k+1

k+1

i

d2

d

k+1

d2

d

k+1

n

i=k+1

k+1

i

ONCLUSION

1ε

CKNOWLEDGMENT

### R

EFERENCES

 A. Ayad, “A lecture on the complexity of factoring polynomials over global fields,” International Mathematical Forum, vol. 5, no. 10, pp.

477–486, 2010.

 J.-F. Biasse and C. Fieker, “A polynomial time algorithm for computing the hnf of a module over the integers of a number field,” 2012, To appear in the proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC).

 D. Boneh, “Finding smooth integers in short intervals using crt decoding,” in Proceedings of the thirty-second annual ACM symposium on Theory of computing, ser. STOC ’00. New York, NY, USA: ACM, 2000, pp. 265–272. [Online]. Available:

http://doi.acm.org/10.1145/335305.335337

 H. Cohen,Advanced topics in computational algebraic number theory, ser. Graduate Texts in Mathematics. Springer-Verlag, 1991, vol. 193.

 H. Cohn and N. Heninger, “Ideal forms of coppersmith’s theorem and guruswami-sudan list decoding,” inProceedings of Innovations in computer science, 2011.

 C. Fieker and D. Stehl´e, “Short bases of lattices over number fields,”

in Algorithmic Number Theory, 9th International Symposium, ANTS- IX, Nancy, France, July 19-23, 2010. Proceedings, ser. Lecture Notes in Computer Science, G. Hanrot, F. Morain, and E. Thom´e, Eds., vol.

6197. Springer, 2010, pp. 157–173.

 O. Goldreich, D. Ron, and M. Sudan, “Chinese remaindering with errors,” in Proceedings of the thirty-first annual ACM symposium on Theory of computing, ser. STOC ’99. New York, NY, USA: ACM, 1999, pp. 225–234.

 V. Guruswami, “Constructions of codes from number fields,” IEEE Transactions on Information Theory, vol. 49, no. 3, pp. 594–603, 2003.

 ——, List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition (Lecture Notes in Computer Science). Secaucus, NJ, USA: Springer-Verlag New York, Inc., 2005.

 V. Guruswami, A. Sahai, and M. Sudan, “Soft-decision decoding of chinese remainder codes,” inProceedings of the 41st Annual Symposium on Foundations of Computer Science. Washington, DC, USA: IEEE Computer Society, 2000, pp. 159–168.

 V. Guruswami and M. Sudan, “Improved decoding of reed-solomon and algebraic-geometric codes,” in IEEE Symposium on Foundations of Computer Science, vol. 5, 1999, pp. 28–39.

 H. Lenstra, “Codes from algebraic number fields,” inMathematics and computer science II, Fundamental contributions in the Netherlands since 1945, ser. CWI Monograph, M. Hazewinkel, J. Lenstra, and L. L.

Meertens, Eds., vol. 4, North-Holland, Amsterdam, 1986, pp. 94–104.

 D. Mandelbaum, “On a class of arithmetic codes and a decoding algorithm (corresp.),”IEEE Transactions on Information Theory, vol. 22, pp. 85–88, 1976.

 J. Neukirch,Algebraic number theory, ser. Comprehensive Studies in Mathematics. Springer-Verlag, 1999, iSBN 3-540-65399-6.

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