May  2017, 11(2): 379-388. doi: 10.3934/amc.2017032

On parameters of subfield subcodes of extended norm-trace codes

1. 

Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico -Río Piedras Campus, San Juan, Puerto Rico, 00925 USA

2. 

Department of Mathematics, University of Puerto Rico -Ponce Campus, Ponce, Puerto Rico, 00716 USA

Received  February 2016 Revised  March 2016 Published  May 2017

In this article we describe how to find the parameters of subfield subcodes of extended Norm-Trace codes (ENT codes). With a Gröbner basis of the ideal of the $\mathbb{F}_{q^r}$ rational points of the extended Norm-Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH-like bound from the minimum distance of the original code. The ENT codes we study here are a more general class of codes than those given in [1]. We study their subfield subcodes as well. We give an example of ENT subfield subcodes that have optimal parameters. Furthermore, we give examples of binary subfield subcodes of ENT codes of very large length for modern applications (e.g. for flash memories).

Citation: Heeralal Janwa, Fernando L. Piñero. On parameters of subfield subcodes of extended norm-trace codes. Advances in Mathematics of Communications, 2017, 11 (2) : 379-388. doi: 10.3934/amc.2017032
References:
[1]

M. Bras-Amor´os and M. E. O'Sullivan, Extended norm-trace codes with optimized correction capability, in Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 2007,337-346. doi: 10.1007/978-3-540-77224-8_39.  Google Scholar

[2] D. A. CoxJ. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An INTRODUCTION to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 2007.  doi: 10.1007/978-0-387-35651-8.  Google Scholar
[3]

J. Fitzgerald and R. F. Lax, Decoding affine variety codes using Gröbner bases, DCC, 13 (1998), 147-158.  doi: 10.1023/A:1008274212057.  Google Scholar

[4]

E. E. Gad, W. Huang, Y. Li and J. Bruck, Rewriting flash memories by message passing, 2016, US Patent App. 15/011,537. Google Scholar

[5]

O. Geil, On codes from norm-trace curves, Finite Fields Appl., 9 (2003), 351-371.  doi: 10.1016/S1071-5797(03)00010-8.  Google Scholar

[6]

O. Geil, Evaluation codes from an affine variety code perspective, in Advances in Algebraic Geometry Codes, World Scientific, 2008. doi: 10.1142/9789812794017_0004.  Google Scholar

[7]

O. Geil and T. Hoholdt, Footprints or generalized bezout's theorem, IEEE Trans. Inf. Theory, 46 (2006), 635-641.  doi: 10.1109/18.825832.  Google Scholar

[8]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes available at http://www.codetables.de Google Scholar

[9]

H. Janwa and F. Piñero, On the parameters of subfield subcodes of norm-trace codes, Congr. Numerant., 206 (2010), 99-113.   Google Scholar

[10]

R. Lidl and H. Niederreiter, Finite fields, in Encyclopaedia of Mathematics and Its Applications, Cambridge Univ. Press, 1997. Google Scholar

[11]

F. L. Piñero and H. Janwa, On the subfield subcodes of Hermitian codes, Des. Codes Crypt., 70 (2014), 157-173.  doi: 10.1007/s10623-012-9736-9.  Google Scholar

[12]

H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inf. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376.  Google Scholar

show all references

References:
[1]

M. Bras-Amor´os and M. E. O'Sullivan, Extended norm-trace codes with optimized correction capability, in Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 2007,337-346. doi: 10.1007/978-3-540-77224-8_39.  Google Scholar

[2] D. A. CoxJ. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An INTRODUCTION to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 2007.  doi: 10.1007/978-0-387-35651-8.  Google Scholar
[3]

J. Fitzgerald and R. F. Lax, Decoding affine variety codes using Gröbner bases, DCC, 13 (1998), 147-158.  doi: 10.1023/A:1008274212057.  Google Scholar

[4]

E. E. Gad, W. Huang, Y. Li and J. Bruck, Rewriting flash memories by message passing, 2016, US Patent App. 15/011,537. Google Scholar

[5]

O. Geil, On codes from norm-trace curves, Finite Fields Appl., 9 (2003), 351-371.  doi: 10.1016/S1071-5797(03)00010-8.  Google Scholar

[6]

O. Geil, Evaluation codes from an affine variety code perspective, in Advances in Algebraic Geometry Codes, World Scientific, 2008. doi: 10.1142/9789812794017_0004.  Google Scholar

[7]

O. Geil and T. Hoholdt, Footprints or generalized bezout's theorem, IEEE Trans. Inf. Theory, 46 (2006), 635-641.  doi: 10.1109/18.825832.  Google Scholar

[8]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes available at http://www.codetables.de Google Scholar

[9]

H. Janwa and F. Piñero, On the parameters of subfield subcodes of norm-trace codes, Congr. Numerant., 206 (2010), 99-113.   Google Scholar

[10]

R. Lidl and H. Niederreiter, Finite fields, in Encyclopaedia of Mathematics and Its Applications, Cambridge Univ. Press, 1997. Google Scholar

[11]

F. L. Piñero and H. Janwa, On the subfield subcodes of Hermitian codes, Des. Codes Crypt., 70 (2014), 157-173.  doi: 10.1007/s10623-012-9736-9.  Google Scholar

[12]

H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inf. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376.  Google Scholar

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