Table 1.
Number of GRS and MDS codes of small length and dimension 3
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On the performance of optimal double circulant even codes
November
2017, 11(4): 777-790.
doi: 10.3934/amc.2017057
Counting generalized Reed-Solomon codes
| 1. | Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, DK-2800 Kgs. Lyngby, Denmark |
| 2. | School of Computer Science, Engineering and Mathematics, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia |
| 3. | Department of Mathematics King Abdulaziz University, Jeddah 21859, Saudi Arabia |
| 4. | Department of Mathematics, Indian Institute of Science Education and Research (IISER), Dr. Homi Bhabha Road, Pashan, Pune 411008, India |
In this article we count the number of $[n, k]$ generalized Reed-Solomon (GRS) codes, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of $[n, 3]$ MDS codes with $n=6, 7, 8, 9$.
Mathematics Subject Classification:
Primary: 94B27, 51E21.
Citation:
Peter Beelen, David Glynn, Tom Høholdt, Krishna Kaipa. Counting generalized Reed-Solomon codes. Advances in Mathematics of Communications,
2017, 11
(4)
: 777-790.
doi: 10.3934/amc.2017057
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References:
| [1] |
S. Ball,
On sets of vectors of a finite vector space in which every subset of basis size is a basis, J. Eur. Math. Soc., 14 (2012), 733-748.
|
| [2] |
S. Ball and J. De Beule,
On sets of vectors of a finite vector space in which every subset of basis size is a basis Ⅱ, Des. Codes Cryptography, 65 (2012), 5-14.
doi: 10.1007/s10623-012-9658-6. |
| [3] |
W.-L. Chow,
On the geometry of algebraic homogeneous spaces, Annals of Mathematics. Second Series, 50 (1949), 32-67.
doi: 10.2307/1969351. |
| [4] |
P. Dembowski, Finite Geometries Springer, 1968. |
| [5] |
A. Dür,
The automorphism groups of Reed-Solomon codes, J. Comb. Theory, Series A, 44 (1987), 69-82.
doi: 10.1016/0097-3165(87)90060-4. |
| [6] |
D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. |
| [7] |
S. R. Ghorpade and G. Lachaud,
Hyperplane sections of Grassmannians and the number of MDS linear codes, Finite Fields and Their Applications, 7 (2001), 468-506.
doi: 10.1006/ffta.2000.0299. |
| [8] |
D. G. Glynn,
The non-classical 10-arc of $\mathrm{PG}(4,9)$, Discrete Mathematics, 59 (1986), 43-51.
|
| [9] |
D. G. Glynn,
Rings of geometries, Ⅱ, J. Comb. Theory, Series A, 49 (1988), 26-66.
doi: 10.1016/0097-3165(88)90027-1. |
| [10] |
D. G. Glynn,
Every oval of $\text{PG}(2,q)$, $q$ even, is the product of its external lines, Bull. Inst. Combin. Appl., 9 (1993), 65-68.
|
| [11] |
D. G. Glynn,
A condition for arcs and MDS codes, Designs Codes and Cryptography, 58 (2011), 215-218.
doi: 10.1007/s10623-010-9404-x. |
| [12] |
D. G. Glynn,
An invariant for hypersurfaces in prime characteristic, SIAM Journal on Discrete Mathematics, 26 (2012), 881-883.
doi: 10.1137/110823274. |
| [13] |
P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978. |
| [14] |
J. Harris, Algebraic Geometry: A First Course, Springer-Verlag, New York, 1995. |
| [15] |
J. W. P. Hirschfeld,
Finite Projective Spaces of Three Dimensions The Clarendon Press, Oxford University Press, New York, 1985. |
| [16] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, second edition, 1998. |
| [17] |
J. W. P. Hirschfeld, G. Korchmaros and F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, NJ, 2008. |
| [18] |
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Springer Monographs in Mathematics, Springer, London, 2016. |
| [19] |
A. V. Iampolskda, A. N. Skorobogatov and E. A. Sorokin,
Formula for the number of $[9,3]$ MDS codes, IEEE Trans. Inf. Theory, 41 (1995), 1667-1671.
doi: 10.1109/18.476239. |
| [20] |
K. V. Kaipa,
An asymptotic formula in $q$ for the number of $[n,k]$ $q$-ary MDS codes, IEEE Trans. Inf. Theory, 60 (2014), 7047-7057.
doi: 10.1109/TIT.2014.2349903. |
| [21] |
R. Rolland,
The number of MDS $[7,3]$ codes on finite fields of characteristic $2$, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 301-310.
doi: 10.1007/BF01294838. |
| [22] |
B. Segre, Lectures on Modern Geometry, Edizioni Cremonese, Roma, 1961. |
| [23] |
R. C. Singleton,
Maximum distance $q$-nary codes, IEEE Trans. Information Theory, 10 (1964), 116-118.
|
show all references
References:
| [1] |
S. Ball,
On sets of vectors of a finite vector space in which every subset of basis size is a basis, J. Eur. Math. Soc., 14 (2012), 733-748.
|
| [2] |
S. Ball and J. De Beule,
On sets of vectors of a finite vector space in which every subset of basis size is a basis Ⅱ, Des. Codes Cryptography, 65 (2012), 5-14.
doi: 10.1007/s10623-012-9658-6. |
| [3] |
W.-L. Chow,
On the geometry of algebraic homogeneous spaces, Annals of Mathematics. Second Series, 50 (1949), 32-67.
doi: 10.2307/1969351. |
| [4] |
P. Dembowski, Finite Geometries Springer, 1968. |
| [5] |
A. Dür,
The automorphism groups of Reed-Solomon codes, J. Comb. Theory, Series A, 44 (1987), 69-82.
doi: 10.1016/0097-3165(87)90060-4. |
| [6] |
D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. |
| [7] |
S. R. Ghorpade and G. Lachaud,
Hyperplane sections of Grassmannians and the number of MDS linear codes, Finite Fields and Their Applications, 7 (2001), 468-506.
doi: 10.1006/ffta.2000.0299. |
| [8] |
D. G. Glynn,
The non-classical 10-arc of $\mathrm{PG}(4,9)$, Discrete Mathematics, 59 (1986), 43-51.
|
| [9] |
D. G. Glynn,
Rings of geometries, Ⅱ, J. Comb. Theory, Series A, 49 (1988), 26-66.
doi: 10.1016/0097-3165(88)90027-1. |
| [10] |
D. G. Glynn,
Every oval of $\text{PG}(2,q)$, $q$ even, is the product of its external lines, Bull. Inst. Combin. Appl., 9 (1993), 65-68.
|
| [11] |
D. G. Glynn,
A condition for arcs and MDS codes, Designs Codes and Cryptography, 58 (2011), 215-218.
doi: 10.1007/s10623-010-9404-x. |
| [12] |
D. G. Glynn,
An invariant for hypersurfaces in prime characteristic, SIAM Journal on Discrete Mathematics, 26 (2012), 881-883.
doi: 10.1137/110823274. |
| [13] |
P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978. |
| [14] |
J. Harris, Algebraic Geometry: A First Course, Springer-Verlag, New York, 1995. |
| [15] |
J. W. P. Hirschfeld,
Finite Projective Spaces of Three Dimensions The Clarendon Press, Oxford University Press, New York, 1985. |
| [16] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, second edition, 1998. |
| [17] |
J. W. P. Hirschfeld, G. Korchmaros and F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, NJ, 2008. |
| [18] |
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Springer Monographs in Mathematics, Springer, London, 2016. |
| [19] |
A. V. Iampolskda, A. N. Skorobogatov and E. A. Sorokin,
Formula for the number of $[9,3]$ MDS codes, IEEE Trans. Inf. Theory, 41 (1995), 1667-1671.
doi: 10.1109/18.476239. |
| [20] |
K. V. Kaipa,
An asymptotic formula in $q$ for the number of $[n,k]$ $q$-ary MDS codes, IEEE Trans. Inf. Theory, 60 (2014), 7047-7057.
doi: 10.1109/TIT.2014.2349903. |
| [21] |
R. Rolland,
The number of MDS $[7,3]$ codes on finite fields of characteristic $2$, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 301-310.
doi: 10.1007/BF01294838. |
| [22] |
B. Segre, Lectures on Modern Geometry, Edizioni Cremonese, Roma, 1961. |
| [23] |
R. C. Singleton,
Maximum distance $q$-nary codes, IEEE Trans. Information Theory, 10 (1964), 116-118.
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