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GHWs of codes derived from the incidence matrices of some graphs

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Abstract

By the rth generalized Hamming weight of a linear code C, denoted by \( d_{r}(C) \), we mean the smallest support size of any r-dimensional subcode of C. In this paper, we determine the rth generalized Hamming weight of the binary linear code C(G) with the parity check matrix A(G) , where the underlying graph G is a complete graph, a complete bipartite graph, a triangular graph or the Kneser graph K(n, 2) , and A(G) is the incidence matrix of G. We also obtain the rth generalized Hamming weight of the dual code of C(G) .

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References

  • Beelen P (2019) A note on the generalized Hamming weights of Reed-Muller codes. Appl Algebra Eng Commun Comput 30(3):233–242

    Article  MathSciNet  Google Scholar 

  • Beelen P, Datta M (2018) Generalized Hamming weights of affine Cartesian codes. Finite Fields Appl 51:130–145

    Article  MathSciNet  Google Scholar 

  • Dankelmann P, Key JD, Rodrigues BG (2013) Codes from incidence matrices of graphs. Des Codes Cryptogr 68(1–3):373–393

    Article  MathSciNet  Google Scholar 

  • Ding C, Yang J (2013) Hamming weights in irreducible cyclic codes. Discrete Math 313(4):434–446

    Article  MathSciNet  Google Scholar 

  • Feng GL, Tzeng KK, Wei VK (1992) On the generalized Hamming weights of several classes of cyclic codes. IEEE Trans Inf Theory 38(3):1125–1130

    Article  MathSciNet  Google Scholar 

  • Fish W, Key JD, Mwambene E (2010) Codes from the incidence matrices and line graphs of Hamming graphs. Discret Math 310:1884–1897

    Article  MathSciNet  Google Scholar 

  • Ghinelli D, Key JD (2011) Codes from incidence matrices and line graphs of Paley graphs. Adv Math Commun 5:93–108

    Article  MathSciNet  Google Scholar 

  • Heijnen P, Pellikaan R (1998) Generalized Hamming weights of q-ary Reed-Muller codes. IEEE Trans Inform Theory 44(1):181–196

    Article  MathSciNet  Google Scholar 

  • Helleseth T, Kløve T, Mykkeltveit J (1977) The weight distribution of irreducible cyclic codes with block lengths \( n_{1}((q^{l}-1)/N) \). Discrete Math 18:179–211

    Article  MathSciNet  Google Scholar 

  • Jian G, Feng R, Wub H (2017) Generalized Hamming weights of three classes of linear codes. Finite Fields Appl 45:341–354

    Article  MathSciNet  Google Scholar 

  • Johnsen T, Verdure H (2017) Generalized Hamming weights for almost affine codes. IEEE Trans Inf Theory 63(4):1941–1953

    Article  MathSciNet  Google Scholar 

  • Key JD, Fish W, Fish E (2011) Codes from the incidence matrices and line graphs of Hamming graphs \( H^{ k}(n, 2) \) for \( k \ge 2 \). Adv Math Commun 5:373–394

    Article  MathSciNet  Google Scholar 

  • Key JD, Moori J, Rodrigues BG (2010) Codes associated with triangular graphs, and permutation decoding. Int J Inform Coding Theory 1(3):334–349

    Article  MathSciNet  Google Scholar 

  • Martínez-Bernal J, Valencia-Bucio MA, Villarreal RH (2020) Generalized Hamming weights of projective Reed–Muller- type codes over graphs. Discrete Math 343:111639

  • Martínez- Bernal J, Valencia-Bucio MA, Villarreal RH (2020) Linear codes over signed graphs. Des Codes Cryptogr 88:273–296

    Article  MathSciNet  Google Scholar 

  • Olaya-León W, Granados-Pinzón C (2015) The second generalized Hamming weight of certain Castle codes. Des Codes Cryptogr 76(1):81–87

  • Schaathun HG, Willems W (2003) A lower bound on the weight hierarchies of product codes. Discrete Appl Math 128(1):251–261

    Article  MathSciNet  Google Scholar 

  • Stichtenoth H, Voss C (1994) Generalized Hamming weights of trace codes. IEEE Trans Inf Theory 40(3):554–558

    Article  MathSciNet  Google Scholar 

  • Tonchev VD (2002) Error-correcting codes from graphs. Discrete Math 257:549–557

    Article  MathSciNet  Google Scholar 

  • Tsfasman MA, Vlădut SG (1995) Geometric approach to higher weights. IEEE Trans Inf Theory 41(6):1564–1588

    Article  MathSciNet  Google Scholar 

  • Wan Z-X (1994) The weight hierarchies of the projective codes from nondegenerate quadrics. Des Codes Cryptogr 4:283–300

    Article  MathSciNet  Google Scholar 

  • Wei VK (1991) Generalized Hamming weights for linear codes. IEEE Trans Inf Theory 37(5):1412–1418

    Article  MathSciNet  Google Scholar 

  • Wei VK, Yang K (1993) On the generalized Hamming weights of product codes. IEEE Trans Inf Theory 39(5):1709–1713

    Article  MathSciNet  Google Scholar 

  • Xiong M, Li S, Ge G (2016) The Weight Hierarchy of Some Reducible Cyclic Codes. IEEE Trans Inf Theory 62(7):4071–4080

    Article  MathSciNet  Google Scholar 

  • Yang M, Li J, Feng K, Lin D (2015) Generalized Hamming weights of irreducible cyclic codes. IEEE Trans Inf Theory 61(9):4905–4913

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their helpful remarks which have improved the presentation of the paper.

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  1. Hamid Reza Maimani and Maryam Mohammadpour Sabet have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, 16785-163, Tehran, Iran

    Hamid Reza Maimani, Maryam Mohammadpour Sabet & Modjtaba Ghorbani

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  1. Hamid Reza Maimani
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  2. Maryam Mohammadpour Sabet
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  3. Modjtaba Ghorbani
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Correspondence to Modjtaba Ghorbani.

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Communicated by Carlos Hoppen.

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Maimani, H.R., Mohammadpour Sabet, M. & Ghorbani, M. GHWs of codes derived from the incidence matrices of some graphs. Comp. Appl. Math. 41, 193 (2022). https://doi.org/10.1007/s40314-022-01891-6

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  • DOI: https://doi.org/10.1007/s40314-022-01891-6

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