American Institute of Mathematical Sciences

Advanced
May  2021, 15(2): 267-277. doi: 10.3934/amc.2020065

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

Nicola Pace 1,, and  Angelo Sonnino 2,

1. 

Chair of Operations Research - Technische Universität München, Arcisstr. 21, 80333 Munich, Germany
2. 

Dipartimento di Matematica, Informatica ed Economia - Università degli Studi della Basilicata, Viale dell'Ateneo Lucano, 10, 85100 Potenza, Italy

* Corresponding author: Nicola Pace

Received  February 2017 Revised  June 2018 Published  January 2020

This paper deals with the problem of determining whether a PD-set exists for a given linear code $ \mathcal C $ and information set $ I $. A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups $ \mathrm{M}_{12} $ and $ \mathrm{M}_{22} $, respectively. In both cases, the existence of a PD–set is proven. In general, the algorithm works well whenever the code $ \mathcal C $ has a very large automorphism group.

Keywords: Automorphism group, linear code, Mathieu group, permutation decoding, transitive code.
Mathematics Subject Classification: Primary: 94B05, 94B35; Secondary: 20B25.
Citation: Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
References:
[1]

R. Abbott, J. Bray, S. Linton, S. Nickerson, S. Norton, R. Parker, I. Suleiman, J. Tripp, P. Walsh and R. Wilson, Atlas of finite group representations - version 3, http://brauer.maths.qmul.ac.uk/Atlas/. Google Scholar

[2]

J. Bierbrauer, S. Marcugini and F. Pambianco, The Pace code, the Mathieu group ${M}_12$ and the small Witt design ${S}(5, 6, 12)$, Discrete Math., 340 (2017), 1187-1190.  doi: 10.1016/j.disc.2016.12.018.  Google Scholar

[3]

W. Bosma, J. J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[4]

M. Braun, A. Kohnert and A. Wassermann, Optimal linear codes from matrix groups, IEEE Trans. Inform. Theory, 51 (2005), 4247-4251.  doi: 10.1109/TIT.2005.859291.  Google Scholar

[5] W. Burnside, Theory of Groups of Finite Order, 2nd edition, Cambridge University Press, Cambridge, 1911.  doi: 10.1017/CBO9781139237253.  Google Scholar
[6]

P. Camion, Linear codes with given automorphism groups, Discrete Math., 3 (1972), 33-45.  doi: 10.1016/0012-365X(72)90023-4.  Google Scholar

[7]

H. Chabanne, Permutation decoding of abelian codes, IEEE Trans. Inform. Theory, 38 (1992), 1826-1829.  doi: 10.1109/18.165460.  Google Scholar

[8]

A. Cossidente, C. Nolè and A. Sonnino, Cap codes arising from duality, Bull. Inst. Combin. Appl., 67 (2013), 33-42.   Google Scholar

[9]

A. Cossidente and A. Sonnino, A geometric construction of a $[110, 5, 90]_9$–linear code admitting the Mathieu group ${M}_11$, IEEE Trans. Inform. Theory, 54 (2008), 5251-5252.  doi: 10.1109/TIT.2008.929966.  Google Scholar

[10]

A. Cossidente and A. Sonnino, Finite geometry and the Gale transform, Discrete Math., 310 (2010), 3206-3210.  doi: 10.1016/j.disc.2009.08.019.  Google Scholar

[11]

A. Cossidente and A. Sonnino, Some recent results in finite geometry and coding theory arising from the Gale transform, Rend. Mat. Appl. (7), 30 (2010), 67-76.   Google Scholar

[12]

A. Cossidente and A. Sonnino, Linear codes arising from the Gale transform of distinguished subsets of some projective spaces, Discrete Math., 312 (2012), 647-651.  doi: 10.1016/j.disc.2011.05.022.  Google Scholar

[13]

A. Cossidente and A. Sonnino, On graphs and codes associated to the sporadic simple groups HS and ${M}_{22}$, Australas. J. Combin., 60 (2014), 208-216.   Google Scholar

[14]

D. Crnković, S. Rukavina and L. Simčić, Binary doubly-even self-dual codes of length 72 with large automorphism groups, Math. Commun., 18 (2013), 297-308.   Google Scholar

[15]

W. Fish, Binary codes and partial permutation decoding sets from the Johnson graphs, Graphs Combin., 31 (2015), 1381-1396.  doi: 10.1007/s00373-014-1485-2.  Google Scholar

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W. Fish, J. D. Key and E. Mwambene, Partial permutation decoding for simplex codes, Adv. Math. Commun., 6 (2012), 505-516.  doi: 10.3934/amc.2012.6.505.  Google Scholar

[17]

W. Fish, K. Kumwenda and E. Mwambene, Codes related to line graphs of triangular graphs and permutation decoding, Quaest. Math., 35 (2012), 489-505.  doi: 10.2989/16073606.2012.742287.  Google Scholar

[18]

M. Giulietti, G. Korchmáros, S. Marcugini and F. Pambianco, Transitive $A_6$-invariant $k$-arcs in $PG(2, q)$, Des. Codes Cryptogr., 68 (2013), 73-79.  doi: 10.1007/s10623-012-9619-0.  Google Scholar

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D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[20]

M. Grassl, Bounds on the Minimum Distance of Linear Codes and Quantum Codes, Online available at http://www.codetables.de. Accessed on January 3, 2020. Google Scholar

[21]

R. Hill, On the largest size of cap in $s_{5, 3}$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 54 (1974), 378-384.   Google Scholar

[22]

R. Hill, A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1986.  Google Scholar

[23]

W. C. Huffman, Codes and groups, in Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), vol. 2, part 2, North-Holland, Amsterdam, 1998, chapter 17, 1345–1440.  Google Scholar

[24]

L. Indaco and G. Korchmáros, 42-arcs in $PG(2, q)$ left invariant by $PSL(2, 7)$, Des. Codes Cryptogr., 64 (2012), 33-46.  doi: 10.1007/s10623-011-9532-y.  Google Scholar

[25]

J. D. Key, Permutation decoding for codes from designs, finite geometries and graphs, in Information Security, Coding Theory and Related Combinatorics (eds. D. Crnkovič and V. Tonchev), vol. 29 of NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 2011,172–201.  Google Scholar

[26]

J. D. Key and J. Limbupasiriporn, Permutation decoding of codes from Paley graphs, Congr. Numer., 170 (2004), 143-155.   Google Scholar

[27]

J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.  doi: 10.1016/j.ejc.2004.04.007.  Google Scholar

[28]

J. D. Key, T. P. McDonough and V. C. Mavron, Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247.  doi: 10.1016/j.ffa.2005.05.007.  Google Scholar

[29]

J. D. Key, J. Moori and B. G. Rodrigues, Permutation decoding for the binary codes from triangular graphs, European J. Combin., 25 (2004), 113-123.  doi: 10.1016/j.ejc.2003.08.001.  Google Scholar

[30]

J. D. Key, J. Moori and B. G. Rodrigues, Binary codes from graphs on triples and permutation decoding, Ars Combin., 79 (2006), 11-19.   Google Scholar

[31]

J. D. Key, J. Moori and B. G. Rodrigues, Partial permutation decoding of some binary codes from graphs on triples, Ars Combin., 91 (2009), 363-371.   Google Scholar

[32]

J. D. Key, J. Moori and B. G. Rodrigues, Codes associated with triangular graphs and permutation decoding, Int. J. Inf. Coding Theory, 1 (2010), 334-349.  doi: 10.1504/IJICOT.2010.032547.  Google Scholar

[33]

J. D. Key and B. G. Rodrigues, Codes from lattice and related graphs, and permutation decoding, Discrete Appl. Math., 158 (2010), 1807-1815.  doi: 10.1016/j.dam.2010.07.003.  Google Scholar

[34]

J. D. Key and P. Seneviratne, Binary codes from rectangular lattice graphs and permutation decoding, European J. Combin., 28 (2007), 121-126.  doi: 10.1016/j.ejc.2005.09.001.  Google Scholar

[35]

J. D. Key and P. Seneviratne, Permutation decoding for binary codes from lattice graphs, Discrete Math., 308 (2008), 2862-2867.  doi: 10.1016/j.disc.2006.06.049.  Google Scholar

[36]

W. Knapp and H.-J. Schaeffer, On the codes related to the Higman-Sims graph, Electron. J. Combin., 22 (2015), Paper 1.19, 58 pp.  Google Scholar

[37]

W. Knapp and P. Schmid, Codes with prescribed permutation group, J. Algebra, 67 (1980), 415-435.  doi: 10.1016/0021-8693(80)90169-6.  Google Scholar

[38]

A. Kohnert, Constructing two-weight codes with prescribed groups of automorphisms, Discrete Appl. Math., 155 (2007), 1451-1457.  doi: 10.1016/j.dam.2007.03.006.  Google Scholar

[39]

A. Kohnert and A. Wassermann, Construction of binary and ternary self-orthogonal linear codes, Discrete Appl. Math., 157 (2009), 2118-2123.  doi: 10.1016/j.dam.2007.10.030.  Google Scholar

[40]

A. Kohnert and J. Zwanzger, New linear codes with prescribed group of automorphisms found by heuristic search, Adv. Math. Commun., 3 (2009), 157-166.  doi: 10.3934/amc.2009.3.157.  Google Scholar

[41]

G. Korchmáros and N. Pace, Infinite family of large complete arcs in $ {\rm{PG}} (2, q^n)$, with $q$ odd and $n>1$ odd, Des. Codes Cryptogr., 55 (2010), 285-296.  doi: 10.1007/s10623-009-9343-6.  Google Scholar

[42]

H.-J. Kroll and R. Vincenti, PD-sets for the codes related to some classical varieties, Discrete Math., 301 (2005), 89-105.  doi: 10.1016/j.disc.2004.11.020.  Google Scholar

[43]

H.-J. Kroll and R. Vincenti, Antiblocking systems and PD-sets, Discrete Math., 308 (2008), 401-407.  doi: 10.1016/j.disc.2006.11.056.  Google Scholar

[44]

H.-J. Kroll and R. Vincenti, PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of $ \rm PG(5, 2)$, Discrete Math., 308 (2008), 408-414.  doi: 10.1016/j.disc.2006.11.057.  Google Scholar

[45]

F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[46]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, North-Holland Mathematical Library, Vol. 16. Google Scholar

[47]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, North-Holland Mathematical Library, Vol. 16.  Google Scholar

[48]

P. M. Neumann, A lemma that is not Burnside's, Math. Sci., 4 (1979), 133-141.   Google Scholar

[49]

N. Pace, New ternary linear codes from projectivity groups, Discrete Math., 331 (2014), 22-26.  doi: 10.1016/j.disc.2014.04.027.  Google Scholar

[50]

N. Pace, On small complete arcs and transitive $A_5$-invariant arcs in the projective plane $PG(2, q)$, J. Combin. Des., 22 (2014), 425-434.  doi: 10.1002/jcd.21372.  Google Scholar

[51]

N. Pace and A. Sonnino, On linear codes admitting large automorphism groups, Des. Codes Cryptogr., 83 (2017), 115-143.  doi: 10.1007/s10623-016-0207-6.  Google Scholar

[52]

B. G. Rodrigues, Self-orthogonal designs and codes from the symplectic groups $s4(3)$ and $s4(4)$, Discrete Math., 308 (2008), 1941-1950.  doi: 10.1016/j.disc.2007.04.047.  Google Scholar

[53]

P. Seneviratne, Codes associated with circulant graphs and permutation decoding, Des. Codes Cryptogr., 70 (2014), 27-33.  doi: 10.1007/s10623-012-9637-y.  Google Scholar

[54]

A. Sonnino, Transitive PSL(2, 7)-invariant 42-arcs in 3-dimensional projective spaces, Des. Codes Cryptogr., 72 (2014), 455-463.  doi: 10.1007/s10623-012-9778-z.  Google Scholar

[55]

B. Stroustrup, The C++ Programming Language, 4th edition, Addison-Wesley, Upper Saddle River, NJ, 2013. Google Scholar

[56]

L. M. G. M. Tolhuizen and W. J. van Gils, A large automorphism group decreases the number of computations in the construction of an optimal encoder/decoder pair for a linear block code, IEEE Trans. Inform. Theory, 34 (1988), 333-338.  doi: 10.1109/18.2646.  Google Scholar

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J. Wolfmann, A permutation decoding of the (24, 12, 8) Golay code, IEEE Trans. Inform. Theory, 29 (1983), 748-750.  doi: 10.1109/TIT.1983.1056726.  Google Scholar

show all references

References:
[1]

R. Abbott, J. Bray, S. Linton, S. Nickerson, S. Norton, R. Parker, I. Suleiman, J. Tripp, P. Walsh and R. Wilson, Atlas of finite group representations - version 3, http://brauer.maths.qmul.ac.uk/Atlas/. Google Scholar

[2]

J. Bierbrauer, S. Marcugini and F. Pambianco, The Pace code, the Mathieu group ${M}_12$ and the small Witt design ${S}(5, 6, 12)$, Discrete Math., 340 (2017), 1187-1190.  doi: 10.1016/j.disc.2016.12.018.  Google Scholar

[3]

W. Bosma, J. J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[4]

M. Braun, A. Kohnert and A. Wassermann, Optimal linear codes from matrix groups, IEEE Trans. Inform. Theory, 51 (2005), 4247-4251.  doi: 10.1109/TIT.2005.859291.  Google Scholar

[5] W. Burnside, Theory of Groups of Finite Order, 2nd edition, Cambridge University Press, Cambridge, 1911.  doi: 10.1017/CBO9781139237253.  Google Scholar
[6]

P. Camion, Linear codes with given automorphism groups, Discrete Math., 3 (1972), 33-45.  doi: 10.1016/0012-365X(72)90023-4.  Google Scholar

[7]

H. Chabanne, Permutation decoding of abelian codes, IEEE Trans. Inform. Theory, 38 (1992), 1826-1829.  doi: 10.1109/18.165460.  Google Scholar

[8]

A. Cossidente, C. Nolè and A. Sonnino, Cap codes arising from duality, Bull. Inst. Combin. Appl., 67 (2013), 33-42.   Google Scholar

[9]

A. Cossidente and A. Sonnino, A geometric construction of a $[110, 5, 90]_9$–linear code admitting the Mathieu group ${M}_11$, IEEE Trans. Inform. Theory, 54 (2008), 5251-5252.  doi: 10.1109/TIT.2008.929966.  Google Scholar

[10]

A. Cossidente and A. Sonnino, Finite geometry and the Gale transform, Discrete Math., 310 (2010), 3206-3210.  doi: 10.1016/j.disc.2009.08.019.  Google Scholar

[11]

A. Cossidente and A. Sonnino, Some recent results in finite geometry and coding theory arising from the Gale transform, Rend. Mat. Appl. (7), 30 (2010), 67-76.   Google Scholar

[12]

A. Cossidente and A. Sonnino, Linear codes arising from the Gale transform of distinguished subsets of some projective spaces, Discrete Math., 312 (2012), 647-651.  doi: 10.1016/j.disc.2011.05.022.  Google Scholar

[13]

A. Cossidente and A. Sonnino, On graphs and codes associated to the sporadic simple groups HS and ${M}_{22}$, Australas. J. Combin., 60 (2014), 208-216.   Google Scholar

[14]

D. Crnković, S. Rukavina and L. Simčić, Binary doubly-even self-dual codes of length 72 with large automorphism groups, Math. Commun., 18 (2013), 297-308.   Google Scholar

[15]

W. Fish, Binary codes and partial permutation decoding sets from the Johnson graphs, Graphs Combin., 31 (2015), 1381-1396.  doi: 10.1007/s00373-014-1485-2.  Google Scholar

[16]

W. Fish, J. D. Key and E. Mwambene, Partial permutation decoding for simplex codes, Adv. Math. Commun., 6 (2012), 505-516.  doi: 10.3934/amc.2012.6.505.  Google Scholar

[17]

W. Fish, K. Kumwenda and E. Mwambene, Codes related to line graphs of triangular graphs and permutation decoding, Quaest. Math., 35 (2012), 489-505.  doi: 10.2989/16073606.2012.742287.  Google Scholar

[18]

M. Giulietti, G. Korchmáros, S. Marcugini and F. Pambianco, Transitive $A_6$-invariant $k$-arcs in $PG(2, q)$, Des. Codes Cryptogr., 68 (2013), 73-79.  doi: 10.1007/s10623-012-9619-0.  Google Scholar

[19]

D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[20]

M. Grassl, Bounds on the Minimum Distance of Linear Codes and Quantum Codes, Online available at http://www.codetables.de. Accessed on January 3, 2020. Google Scholar

[21]

R. Hill, On the largest size of cap in $s_{5, 3}$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 54 (1974), 378-384.   Google Scholar

[22]

R. Hill, A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1986.  Google Scholar

[23]

W. C. Huffman, Codes and groups, in Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), vol. 2, part 2, North-Holland, Amsterdam, 1998, chapter 17, 1345–1440.  Google Scholar

[24]

L. Indaco and G. Korchmáros, 42-arcs in $PG(2, q)$ left invariant by $PSL(2, 7)$, Des. Codes Cryptogr., 64 (2012), 33-46.  doi: 10.1007/s10623-011-9532-y.  Google Scholar

[25]

J. D. Key, Permutation decoding for codes from designs, finite geometries and graphs, in Information Security, Coding Theory and Related Combinatorics (eds. D. Crnkovič and V. Tonchev), vol. 29 of NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 2011,172–201.  Google Scholar

[26]

J. D. Key and J. Limbupasiriporn, Permutation decoding of codes from Paley graphs, Congr. Numer., 170 (2004), 143-155.   Google Scholar

[27]

J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.  doi: 10.1016/j.ejc.2004.04.007.  Google Scholar

[28]

J. D. Key, T. P. McDonough and V. C. Mavron, Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247.  doi: 10.1016/j.ffa.2005.05.007.  Google Scholar

[29]

J. D. Key, J. Moori and B. G. Rodrigues, Permutation decoding for the binary codes from triangular graphs, European J. Combin., 25 (2004), 113-123.  doi: 10.1016/j.ejc.2003.08.001.  Google Scholar

[30]

J. D. Key, J. Moori and B. G. Rodrigues, Binary codes from graphs on triples and permutation decoding, Ars Combin., 79 (2006), 11-19.   Google Scholar

[31]

J. D. Key, J. Moori and B. G. Rodrigues, Partial permutation decoding of some binary codes from graphs on triples, Ars Combin., 91 (2009), 363-371.   Google Scholar

[32]

J. D. Key, J. Moori and B. G. Rodrigues, Codes associated with triangular graphs and permutation decoding, Int. J. Inf. Coding Theory, 1 (2010), 334-349.  doi: 10.1504/IJICOT.2010.032547.  Google Scholar

[33]

J. D. Key and B. G. Rodrigues, Codes from lattice and related graphs, and permutation decoding, Discrete Appl. Math., 158 (2010), 1807-1815.  doi: 10.1016/j.dam.2010.07.003.  Google Scholar

[34]

J. D. Key and P. Seneviratne, Binary codes from rectangular lattice graphs and permutation decoding, European J. Combin., 28 (2007), 121-126.  doi: 10.1016/j.ejc.2005.09.001.  Google Scholar

[35]

J. D. Key and P. Seneviratne, Permutation decoding for binary codes from lattice graphs, Discrete Math., 308 (2008), 2862-2867.  doi: 10.1016/j.disc.2006.06.049.  Google Scholar

[36]

W. Knapp and H.-J. Schaeffer, On the codes related to the Higman-Sims graph, Electron. J. Combin., 22 (2015), Paper 1.19, 58 pp.  Google Scholar

[37]

W. Knapp and P. Schmid, Codes with prescribed permutation group, J. Algebra, 67 (1980), 415-435.  doi: 10.1016/0021-8693(80)90169-6.  Google Scholar

[38]

A. Kohnert, Constructing two-weight codes with prescribed groups of automorphisms, Discrete Appl. Math., 155 (2007), 1451-1457.  doi: 10.1016/j.dam.2007.03.006.  Google Scholar

[39]

A. Kohnert and A. Wassermann, Construction of binary and ternary self-orthogonal linear codes, Discrete Appl. Math., 157 (2009), 2118-2123.  doi: 10.1016/j.dam.2007.10.030.  Google Scholar

[40]

A. Kohnert and J. Zwanzger, New linear codes with prescribed group of automorphisms found by heuristic search, Adv. Math. Commun., 3 (2009), 157-166.  doi: 10.3934/amc.2009.3.157.  Google Scholar

[41]

G. Korchmáros and N. Pace, Infinite family of large complete arcs in $ {\rm{PG}} (2, q^n)$, with $q$ odd and $n>1$ odd, Des. Codes Cryptogr., 55 (2010), 285-296.  doi: 10.1007/s10623-009-9343-6.  Google Scholar

[42]

H.-J. Kroll and R. Vincenti, PD-sets for the codes related to some classical varieties, Discrete Math., 301 (2005), 89-105.  doi: 10.1016/j.disc.2004.11.020.  Google Scholar

[43]

H.-J. Kroll and R. Vincenti, Antiblocking systems and PD-sets, Discrete Math., 308 (2008), 401-407.  doi: 10.1016/j.disc.2006.11.056.  Google Scholar

[44]

H.-J. Kroll and R. Vincenti, PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of $ \rm PG(5, 2)$, Discrete Math., 308 (2008), 408-414.  doi: 10.1016/j.disc.2006.11.057.  Google Scholar

[45]

F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[46]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, North-Holland Mathematical Library, Vol. 16. Google Scholar

[47]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, North-Holland Mathematical Library, Vol. 16.  Google Scholar

[48]

P. M. Neumann, A lemma that is not Burnside's, Math. Sci., 4 (1979), 133-141.   Google Scholar

[49]

N. Pace, New ternary linear codes from projectivity groups, Discrete Math., 331 (2014), 22-26.  doi: 10.1016/j.disc.2014.04.027.  Google Scholar

[50]

N. Pace, On small complete arcs and transitive $A_5$-invariant arcs in the projective plane $PG(2, q)$, J. Combin. Des., 22 (2014), 425-434.  doi: 10.1002/jcd.21372.  Google Scholar

[51]

N. Pace and A. Sonnino, On linear codes admitting large automorphism groups, Des. Codes Cryptogr., 83 (2017), 115-143.  doi: 10.1007/s10623-016-0207-6.  Google Scholar

[52]

B. G. Rodrigues, Self-orthogonal designs and codes from the symplectic groups $s4(3)$ and $s4(4)$, Discrete Math., 308 (2008), 1941-1950.  doi: 10.1016/j.disc.2007.04.047.  Google Scholar

[53]

P. Seneviratne, Codes associated with circulant graphs and permutation decoding, Des. Codes Cryptogr., 70 (2014), 27-33.  doi: 10.1007/s10623-012-9637-y.  Google Scholar

[54]

A. Sonnino, Transitive PSL(2, 7)-invariant 42-arcs in 3-dimensional projective spaces, Des. Codes Cryptogr., 72 (2014), 455-463.  doi: 10.1007/s10623-012-9778-z.  Google Scholar

[55]

B. Stroustrup, The C++ Programming Language, 4th edition, Addison-Wesley, Upper Saddle River, NJ, 2013. Google Scholar

[56]

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