|
[1]
|
E. F. Assmus Jr. and J. D. Key, Designs and their Codes, Cambridge Tracts in Math., 103, Cambridge Univ. Press, 1992.
doi: 10.1017/CBO9781316529836.
|
|
[2]
|
S. Ballet and R. Rolland, On low weight codewords of generalized affine and projective Reed-Muller codes, Des. Codes Cryptogr., 73 (2014), 271-297.
doi: 10.1007/s10623-013-9911-7.
|
|
[3]
|
P. Beelen and M. Datta, Generalized Hamming weights of affine Cartesian codes, Finite Fields Appl., 51 (2018), 130-145.
doi: 10.1016/j.ffa.2018.01.006.
|
|
[4]
|
P. Beelen, M. Datta and S. R. Ghorpade, Vanishing ideals of projective spaces over finite fields and a projective footprint bound, Acta Math. Sin. (Engl. Ser.), 35 (2019), 47-63.
doi: 10.1007/s10114-018-8024-7.
|
|
[5]
|
P. Beelen, M. Datta and S. R. Ghorpade, A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields, Moscow Math J., 22 (2022), 565-593.
doi: 10.17323/1609-4514-2022-22-4-565-593.
|
|
[6]
|
T. Berger, Automorphism groups of homogeneous and projective Reed-Muller codes, IEEE Trans. Inform. Theory, 48 (2002), 1035-1045.
doi: 10.1109/18.995540.
|
|
[7]
|
T. Berger and P. Charpin, The automorphism group of generalized Reed-Muller codes, Discrete Math., 117 (1993), 1-17.
doi: 10.1016/0012-365X(93)90321-J.
|
|
[8]
|
M. B. Can, R. Joshua and G. V. Ravindra, Higher Grassmann codes Ⅱ, Finite Fields Appl., 89 (2023), Art. 102211, 21 pp.
doi: 10.1016/j.ffa.2023.102211.
|
|
[9]
|
C. Carvalho, V. G. L. Neumann and H. H. Lopez, Projective nested cartesian codes, Bull. Braz. Math. Soc., 48 (2017), 283-302.
doi: 10.1007/s00574-016-0010-z.
|
|
[10]
|
P. Delsarte, J. M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Information and Control, 16 (1970), 403-442.
doi: 10.1016/S0019-9958(70)90214-7.
|
|
[11]
|
S. R. Ghorpade, A note on Nullstellensatz over finite fields, in Contributions in Algebra and Algebraic Geometry (Aurangabad, 2017), Contemp. Math., 738, Amer. Math. Soc., Providence, 2019, 23-32.
doi: 10.1090/conm/738/14876.
|
|
[12]
|
S. R. Ghorpade and R. Ludhani, On the purity of resolutions of Stanley-Reisner rings associated to Reed-Muller codes, in: Algebra and Related Topics with Applications (Aligarh, 2019), Springer Proc. Math. Stat., 392, Springer, Singapore, 2022, 325-335.
doi: 10.1007/978-981-19-3898-6_26.
|
|
[13]
|
S. R. Ghorpade, C. Ritzenthaler, F. Rodier and M. A. Tsfasman, Arithmetic, geometry, and coding theory: Homage to Gilles Lachaud, Arithmetic, Geometry, Cryptography and Coding Theory, Contemp. Math., 770, Amer. Math. Soc., Providence, 2021, 131-150.
doi: 10.1090/conm/770/15433.
|
|
[14]
|
P. Heijnen and R. Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes, IEEE Trans. Inform. Theory, 44 (1998), 181-196.
doi: 10.1109/18.651015.
|
|
[15]
|
N. Kaplan and V. Matei, Counting plane cubic curves over finite fields with a prescribed number of rational intersection points, Eur. J. Math., 7 (2021), 1137-1181.
doi: 10.1007/s40879-021-00472-x.
|
|
[16]
|
T. Kasami, S. Lin and W. W. Peterson, New generalization of the Reed-Muller codes–Part Ⅰ: Primitive Codes, IEEE Trans. Inform. Theory, IT-14 (1968), 189-199.
doi: 10.1109/tit.1968.1054127.
|
|
[17]
|
G. Lachaud, Projective Reed-Muller codes, in Coding Theory and Applications (Cachan, 1986), Lecture Notes in Comput. Sci., 311, Springer, Berlin, 1988, 125-129.
doi: 10.1007/3-540-19368-5_13.
|
|
[18]
|
G. Lachaud, The parameters of projective Reed-Muller codes, Discrete Math., 81 (1990), 217-221.
doi: 10.1016/0012-365X(90)90155-B.
|
|
[19]
|
E. Leducq, A new proof of Delsarte, Goethals and MacWilliams theorem on minimal weight codewords of generalized Reed-Muller codes, Finite Fields Appl., 18 (2012), 581-586.
doi: 10.1016/j.ffa.2011.12.003.
|
|
[20]
|
D. J. Mercier and R. Rolland, Polynômes homogènes qui s$'$annulent sur l$'$espace projectif ${{\mathbb P}}^m({{\mathbb F}_q})$, J. Pure Appl. Algebra, 124 (1998), 227-240.
doi: 10.1016/S0022-4049(96)00104-1.
|
|
[21]
|
D. E. Muller, Application of Boolean algebra to switching circuit design and to error detection, IRE Trans. Electron. Comput., EC-3 (1954), 6-12.
|
|
[22]
|
I. S. Reed, A class of multiple-error-correcting codes and the decoding scheme, Trans. IRE, (1954), 38-49.
|
|
[23]
|
I. S. Reed, A brief history of the development of error correcting codes, Comput. Math. Appl., 39 (2000), 89-93.
doi: 10.1016/S0898-1221(00)00112-7.
|
|
[24]
|
C. Rentería and H. Tapia-Recillas, Reed-muller codes: An ideal theory approach, Comm. Algebra, 25 (1997), 401-413.
doi: 10.1080/00927879708825862.
|
|
[25]
|
R. Rolland, Number of points of non-absolutely irreducible hypersurfaces, in Algebraic Geometry and its Applications (Papette, 2007), Ser. Number Theory Appl., 5, World Scientific, Singapore, 2008, 481-487.
doi: 10.1142/9789812793430_0026.
|
|
[26]
|
J.-P. Serre, Lettre à M. Tsfasman, in Journées Arithmétiques (Luminy, 1989), Astérisque, 198-199-200 (1991), 351-353.
|
|
[27]
|
A. B. Sørensen, Projective Reed-Muller codes, IEEE Trans. Inform. Theory, 37 (1991), 1567-1576.
doi: 10.1109/18.104317.
|
|
[28]
|
S. G. Vlèduts and Yu. I. Manin, Linear codes and modular curves, J. Sov. Math., 30 (1985), 2611-2643.
doi: 10.1007/BF02249124.
|
{{journal.titleEn}} 
DownLoad: