# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021024
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## On additive MDS codes over small fields

 1 Departament de Matemàtiques, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain 2 Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul, Turkey

* Corresponding author: Simeon Ball

Received  December 2020 Revised  April 2021 Early access July 2021

Fund Project: The first author acknowledges the support of the project MTM2017-82166-P of the Spanish Ministerio de Ciencia y Innovación

Let $C$ be a $(n,q^{2k},n-k+1)_{q^2}$ additive MDS code which is linear over ${\mathbb F}_q$. We prove that if $n \geq q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ is linear over ${\mathbb F}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb F}_q$ for $q \in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb F}_{16}$ which are linear over ${\mathbb F}_4$. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $q \in \{ 2,3\}$.

Citation: Simeon Ball, Guillermo Gamboa, Michel Lavrauw. On additive MDS codes over small fields. Advances in Mathematics of Communications, doi: 10.3934/amc.2021024
##### References:
 [1] T. L. Alderson, $(6, 3)$-MDS codes over an alphabet of size $4$, Des. Codes Cryptogr, 38 (2006), 31–40. doi: 10.1007/s10623-004-5659-4.  Google Scholar [2] 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. doi: 10.4171/JEMS/316.  Google Scholar [3] S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133–172. doi: 10.4171/emss/33.  Google Scholar [4] A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, Error-Correcting Linear Codes. Classification by Isometry and Applications, Algorithms and Computation in Mathematics 18, Springer, 2006.  Google Scholar [5] A. Blokhuis and A. E. Brouwer, Small additive quaternary codes, European J. Combin., 25 (2004), 161–167. doi: 10.1016/S0195-6698(03)00096-9.  Google Scholar [6] K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes, Inform and Control, 37 (1978), 19–22. doi: 10.1016/S0019-9958(78)90389-3.  Google Scholar [7] K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Statistics, 23 (1952), 426–434. doi: 10.1214/aoms/1177729387.  Google Scholar [8] P. Dembowski, Finite Geometries, Reprint of the 1968 original. Classics in Mathematics. Springer-Verlag, Berlin, 1997.  Google Scholar [9] J. Bamberg, A. Betten, Ph. Cara, J. De Beule, M. Lavrauw and M. Neunhöffer, Finite Incidence Geometry, FinInG–a GAP Package, Version 1.4.1, 2018. https://www.gap-system.org/Packages/fining.html. Google Scholar [10] G. A. Gamboa Quintero, Additive MDS codes, Master's Thesis, Universitat Politècnica Catalunya, 2020. Google Scholar [11] The GAP Group, GAP – Groups, Algorithms, Programming -a System for Computational Discrete Algebra, Version 4.11.0, 2020. https://www.gap-system.org. Google Scholar [12] L. H. Soicher, GAP Package GRAPE, Version 4.8.5, 2021. https://gap-packages.github.io/grape. Google Scholar [13] M. Grassl and M. Rötteler, Quantum MDS codes over small fields, in Proc. Int. Symp. Inf. Theory (ISIT), (2015), 1104–1108, arXiv: 1502.05267. doi: 10.1109/ISIT.2015.7282626.  Google Scholar [14] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: Update 2001, Finite Geometries, Dev. Math., Kluwer Acad. Publ, Dordrecht, 3 (2001), 201-246.  doi: 10.1007/978-1-4613-0283-4_13.  Google Scholar [15] F. Huber and M. Grassl, Quantum codes of maximal distance and highly entangled subspaces, Quantum, 4 (2020), 284, arXiv: 1907.07733. doi: 10.22331/q-2020-06-18-284.  Google Scholar [16] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.  Google Scholar [17] J. I. Kokkala, D. S. Krotov and P. R. J. Östergård, On the classification of MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 6485–6492. doi: 10.1109/TIT.2015.2488659.  Google Scholar [18] J. I. Kokkala and P. R. J. Östergård, Further results on the classification of MDS codes, Adv. Math. Commun., 10 (2016), 489–498. doi: 10.3934/amc.2016020.  Google Scholar [19] M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, in: Contemporary Mathematics, (eds: G Kyureghyan, GL Mullen, and A Pott), American Mathematical Society, 632 (2015), 271–293. doi: 10.1090/conm/632/12633.  Google Scholar [20] S. Linton, Finding the smallest image of a set, in: ISSAC '04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, 2004 (2004), 229–234. doi: 10.1145/1005285.1005319.  Google Scholar [21] L. Lunelli and M. Sce, Considerazione aritmetiche e risultati sperimentali sui $\{K; n\}_q$-archi, Ist. Lombardo Accad. Sci. Rend. A, 98 (1964), 3-52.   Google Scholar [22] F. J. MacWilliams, Combinatorial Problems of Elementary Abelian Groups, Thesis (Ph.D.)–Radcliffe College, 1962.  Google Scholar [23] K. Shiromoto, Note on MDS codes over the integers modulo $p^{m}$,, Hokkaido Mathematical Journal, 29 (2000), 149–157. doi: 10.14492/hokmj/1350912961.  Google Scholar [24] L. H. Soicher, Computation of partial spreads web-page, http://www.maths.qmul.ac.uk/~lsoicher/partialspreads/ Google Scholar [25] H. N. Ward and J. A. Wood, Characters and the equivalence of codes, J. Combin. Theory Ser. A, 73 (1996), 348–352. doi: 10.1016/S0097-3165(96)80011-2.  Google Scholar

show all references

##### References:
 [1] T. L. Alderson, $(6, 3)$-MDS codes over an alphabet of size $4$, Des. Codes Cryptogr, 38 (2006), 31–40. doi: 10.1007/s10623-004-5659-4.  Google Scholar [2] 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. doi: 10.4171/JEMS/316.  Google Scholar [3] S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133–172. doi: 10.4171/emss/33.  Google Scholar [4] A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, Error-Correcting Linear Codes. Classification by Isometry and Applications, Algorithms and Computation in Mathematics 18, Springer, 2006.  Google Scholar [5] A. Blokhuis and A. E. Brouwer, Small additive quaternary codes, European J. Combin., 25 (2004), 161–167. doi: 10.1016/S0195-6698(03)00096-9.  Google Scholar [6] K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes, Inform and Control, 37 (1978), 19–22. doi: 10.1016/S0019-9958(78)90389-3.  Google Scholar [7] K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Statistics, 23 (1952), 426–434. doi: 10.1214/aoms/1177729387.  Google Scholar [8] P. Dembowski, Finite Geometries, Reprint of the 1968 original. Classics in Mathematics. Springer-Verlag, Berlin, 1997.  Google Scholar [9] J. Bamberg, A. Betten, Ph. Cara, J. De Beule, M. Lavrauw and M. Neunhöffer, Finite Incidence Geometry, FinInG–a GAP Package, Version 1.4.1, 2018. https://www.gap-system.org/Packages/fining.html. Google Scholar [10] G. A. Gamboa Quintero, Additive MDS codes, Master's Thesis, Universitat Politècnica Catalunya, 2020. Google Scholar [11] The GAP Group, GAP – Groups, Algorithms, Programming -a System for Computational Discrete Algebra, Version 4.11.0, 2020. https://www.gap-system.org. Google Scholar [12] L. H. Soicher, GAP Package GRAPE, Version 4.8.5, 2021. https://gap-packages.github.io/grape. Google Scholar [13] M. Grassl and M. Rötteler, Quantum MDS codes over small fields, in Proc. Int. Symp. Inf. Theory (ISIT), (2015), 1104–1108, arXiv: 1502.05267. doi: 10.1109/ISIT.2015.7282626.  Google Scholar [14] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: Update 2001, Finite Geometries, Dev. Math., Kluwer Acad. Publ, Dordrecht, 3 (2001), 201-246.  doi: 10.1007/978-1-4613-0283-4_13.  Google Scholar [15] F. Huber and M. Grassl, Quantum codes of maximal distance and highly entangled subspaces, Quantum, 4 (2020), 284, arXiv: 1907.07733. doi: 10.22331/q-2020-06-18-284.  Google Scholar [16] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.  Google Scholar [17] J. I. Kokkala, D. S. Krotov and P. R. J. Östergård, On the classification of MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 6485–6492. doi: 10.1109/TIT.2015.2488659.  Google Scholar [18] J. I. Kokkala and P. R. J. Östergård, Further results on the classification of MDS codes, Adv. Math. Commun., 10 (2016), 489–498. doi: 10.3934/amc.2016020.  Google Scholar [19] M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, in: Contemporary Mathematics, (eds: G Kyureghyan, GL Mullen, and A Pott), American Mathematical Society, 632 (2015), 271–293. doi: 10.1090/conm/632/12633.  Google Scholar [20] S. Linton, Finding the smallest image of a set, in: ISSAC '04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, 2004 (2004), 229–234. doi: 10.1145/1005285.1005319.  Google Scholar [21] L. Lunelli and M. Sce, Considerazione aritmetiche e risultati sperimentali sui $\{K; n\}_q$-archi, Ist. Lombardo Accad. Sci. Rend. A, 98 (1964), 3-52.   Google Scholar [22] F. J. MacWilliams, Combinatorial Problems of Elementary Abelian Groups, Thesis (Ph.D.)–Radcliffe College, 1962.  Google Scholar [23] K. Shiromoto, Note on MDS codes over the integers modulo $p^{m}$,, Hokkaido Mathematical Journal, 29 (2000), 149–157. doi: 10.14492/hokmj/1350912961.  Google Scholar [24] L. H. Soicher, Computation of partial spreads web-page, http://www.maths.qmul.ac.uk/~lsoicher/partialspreads/ Google Scholar [25] H. N. Ward and J. A. Wood, Characters and the equivalence of codes, J. Combin. Theory Ser. A, 73 (1996), 348–352. doi: 10.1016/S0097-3165(96)80011-2.  Google Scholar
The classification of arcs of lines of $\mathrm{PG}(5, 2)$
 size 4 5 6 number of arcs of points of $\mathrm{PG}(2, 4)$ 1 1 1 number of arcs of lines of $\mathrm{PG}(5, 2)$ 1 1 1
 size 4 5 6 number of arcs of points of $\mathrm{PG}(2, 4)$ 1 1 1 number of arcs of lines of $\mathrm{PG}(5, 2)$ 1 1 1
The classification of arcs of planes of PG(8, 2)
 size 4 5 6 7 8 9 10 number of arcs of points of PG(2, 8) 1 1 3 2 2 2 1 number of arcs of planes of PG(8, 2) 1 2 4 2 2 2 1
 size 4 5 6 7 8 9 10 number of arcs of points of PG(2, 8) 1 1 3 2 2 2 1 number of arcs of planes of PG(8, 2) 1 2 4 2 2 2 1
The classification of arcs of lines of PG(5, 3)
 size 4 5 6 7 8 9 10 # of arcs of points of PG(2, 9) 1 2 6 3 2 1 1 # of arcs of lines of PG(5, 3) 1 4 13 4 3 1 1
 size 4 5 6 7 8 9 10 # of arcs of points of PG(2, 9) 1 2 6 3 2 1 1 # of arcs of lines of PG(5, 3) 1 4 13 4 3 1 1
The classification of arcs of lines of PG(3, 3)
 size 4 5 6 7 8 9 10 # of arcs of points of PG(1, 9) 2 2 2 1 1 1 1 # of arcs of lines of PG(3, 3) 3 4 5 4 3 2 2
 size 4 5 6 7 8 9 10 # of arcs of points of PG(1, 9) 2 2 2 1 1 1 1 # of arcs of lines of PG(3, 3) 3 4 5 4 3 2 2
The classification of arcs of lines of $\mathrm{PG}(5, 4)$
 size 5 6 7 8 9 10 11 # of arcs of $\mathrm{PG}(2, 16)$ 3 22 125 865 1534 1262 300 # of line-arcs of $\mathrm{PG}(5, 4)$ 10 360 8294 15162 2869 1465 301 size 12 13 14 15 16 17 18 # of arcs of $\mathrm{PG}(2, 16)$ 159 70 30 9 5 3 2 # of line-arcs of $\mathrm{PG}(5, 4)$ 159 70 30 9 5 3 2
 size 5 6 7 8 9 10 11 # of arcs of $\mathrm{PG}(2, 16)$ 3 22 125 865 1534 1262 300 # of line-arcs of $\mathrm{PG}(5, 4)$ 10 360 8294 15162 2869 1465 301 size 12 13 14 15 16 17 18 # of arcs of $\mathrm{PG}(2, 16)$ 159 70 30 9 5 3 2 # of line-arcs of $\mathrm{PG}(5, 4)$ 159 70 30 9 5 3 2
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