2000 character limit reached
A recursive construction for projective Reed-Muller codes (2312.05072v2)
Published 8 Dec 2023 in cs.IT, math.AC, and math.IT
Abstract: We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked.
- Maximum number of common zeros of homogeneous polynomials over finite fields. Proc. Amer. Math. Soc., 146(4):1451–1468, 2018.
- Vanishing ideals of projective spaces over finite fields and a projective footprint bound. Acta Math. Sin. (Engl. Ser.), 35(1):47–63, 2019.
- A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields. Mosc. Math. J., 22(4):565–593, 2022.
- J. Bierbrauer. The theory of cyclic codes and a generalization to additive codes. Des. Codes Cryptogr., 25(2):189–206, 2002.
- T. Blackmore and G. H. Norton. Matrix-product codes over 𝔽qsubscript𝔽𝑞\mathbb{F}_{q}blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. Appl. Algebra Engrg. Comm. Comput., 12(6):477–500, 2001.
- M. Boguslavsky. On the number of solutions of polynomial systems. Finite Fields Appl., 3(4):287–299, 1997.
- The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
- M. Datta and S. R. Ghorpade. Number of solutions of systems of homogeneous polynomial equations over finite fields. Proc. Amer. Math. Soc., 145(2):525–541, 2017.
- On generalized Reed-Muller codes and their relatives. Information and Control, 16:403–442, 1970.
- New binary and ternary LCD codes. IEEE Trans. Inform. Theory, 65(2):1008–1016, 2019.
- C. Galindo and F. Hernando. Quantum codes from affine variety codes and their subfield-subcodes. Des. Codes Cryptogr., 76(1):89–100, 2015.
- Stabilizer quantum codes from J𝐽Jitalic_J-affine variety codes and a new Steane-like enlargement. Quantum Inf. Process., 14(9):3211–3231, 2015.
- Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes. ArXiv 2304.08121, 2023.
- Subfield subcodes of projective Reed-Muller codes. ArXiv 2307.09298, 2023.
- M. Grassl. Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de, 2007. Accessed on 2023-04-04.
- P. Heijnen and R. Pellikaan. Generalized Hamming weights of q𝑞qitalic_q-ary Reed-Muller codes. IEEE Trans. Inform. Theory, 44(1):181–196, 1998.
- New generalizations of the Reed-Muller codes. I. Primitive codes. IEEE Trans. Inform. Theory, IT-14:189–199, 1968.
- G. Lachaud. Projective Reed-Muller codes. In Coding theory and applications (Cachan, 1986), volume 311 of Lecture Notes in Comput. Sci., pages 125–129. Springer, Berlin, 1988.
- C. Rentería and H. Tapia-Recillas. Reed-Muller codes: an ideal theory approach. Comm. Algebra, 25(2):401–413, 1997.
- A. B. Sørensen. Projective Reed-Muller codes. IEEE Trans. Inform. Theory, 37(6):1567–1576, 1991.
- V. K. Wei. Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory, 37(5):1412–1418, 1991.