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On the fourth weight of generalized Reed–Muller codes

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Abstract

In this paper, we study the fourth weight of generalized Reed–Muller codes. Erickson in his Ph.D. thesis proved that the second weight of \(R_q(a(q-1)+b,m)\) depends on the second weight \(R_q(b,2)\). Also, Leducq (Discret Math 338:1515–1535, 2015) proved that under the same condition, by the third weight of \(R_q(b,2)\) we can determine the third weight of \(R_q(a(q-1)+b,m)\). In this paper we will show that the similar result does not hold for the fourth weight of generalized Reed–Muller codes. We will determine the fourth weight of generalized Reed–Muller codes of order \(r=a(q-1)+b\) with \(3\le b< \frac{q+4}{3}\).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Somayyeh Golalizadeh & Nasrin Soltankhah

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  1. Somayyeh Golalizadeh
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  2. Nasrin Soltankhah
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Correspondence to Nasrin Soltankhah.

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Communicated by V. A. Zinoviev.

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Golalizadeh, S., Soltankhah, N. On the fourth weight of generalized Reed–Muller codes. Des. Codes Cryptogr. 91, 3857–3879 (2023). https://doi.org/10.1007/s10623-023-01276-8

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  • DOI: https://doi.org/10.1007/s10623-023-01276-8

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