May  2011, 5(2): 395-406. doi: 10.3934/amc.2011.5.395

On the weight distribution of codes over finite rings

1. 

School of Mathematical Sciences, University College Dublin, Springfield, MO 65801-2604, United States

Received  May 2010 Revised  November 2010 Published  May 2011

Let $R>S$ be finite Frobenius rings for which there exists a trace map $T:$ S$R \rightarrow$S$R$. Let $C$f,s$:=\{x \mapsto T(\alpha x + \beta f(x)) : \alpha, \beta \in R \}$. $C$f,s is an $S$-linear subring-subcode of a left linear code over $R$. We consider functions $f$ for which the homogeneous weight distribution of $C$f,s can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.
Citation: Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
References:
[1]

C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of almost perfect nonlinear trinomials and multinomials,, Finite Fields Appl., 14 (2008), 703.  doi: 10.1016/j.ffa.2007.11.002.  Google Scholar

[2]

E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs,, Des. Codes Crypt., 48 (2008), 1.  doi: 10.1007/s10623-007-9136-8.  Google Scholar

[3]

E. Byrne, M. Greferath, A. Kohnert and V. Skachek, New bounds for codes over finite Frobenius rings,, Des. Codes Crypt., 57 (2010), 169.  doi: 10.1007/s10623-009-9359-y.  Google Scholar

[4]

E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings,, Des. Codes Crypt., 42 (2007), 289.  doi: 10.1007/s10623-006-9035-4.  Google Scholar

[5]

E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings,, in, (2010).   Google Scholar

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125.  doi: 10.1023/A:1008344232130.  Google Scholar

[7]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes,, IEEE Trans. Inform. Theory, 51 (2005), 2089.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[8]

I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers (in Russian),, Problemy Peredachi Informatsii, 33 (1997), 22.   Google Scholar

[9]

P. Delsarte, Weights of linear codes and strongly regular normed spaces,, Discrete Math., 3 (1972), 47.  doi: 10.1016/0012-365X(72)90024-6.  Google Scholar

[10]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.  doi: 10.1142/S0219498804000873.  Google Scholar

[11]

M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights,, Discrete Math., 289 (2004), 11.  doi: 10.1016/j.disc.2004.10.002.  Google Scholar

[12]

M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[13]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[14]

R. C. Heimiller, Phase shift pulse codes with good periodic correlation properties,, IRE Trans. Inform. Theory, IT-7 (1961), 254.  doi: 10.1109/TIT.1961.1057655.  Google Scholar

[15]

T. Honold, Characterization of finite Frobenius rings,, Arch. Math. (Basel), 76 (2001), 406.   Google Scholar

[16]

T. Honold, Further results on homogeneous two-weight codes,, in, (2007).   Google Scholar

[17]

T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999).   Google Scholar

[18]

B. R. McDonald, Finite rings with identity,, in, (1974).   Google Scholar

[19]

R. Raghavendran, Finite associative rings,, Compositio Math., 21 (1969), 195.   Google Scholar

[20]

J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions,, IEEE Trans. Inform. Theory, 52 (2006), 712.  doi: 10.1109/TIT.2005.862125.  Google Scholar

show all references

References:
[1]

C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of almost perfect nonlinear trinomials and multinomials,, Finite Fields Appl., 14 (2008), 703.  doi: 10.1016/j.ffa.2007.11.002.  Google Scholar

[2]

E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs,, Des. Codes Crypt., 48 (2008), 1.  doi: 10.1007/s10623-007-9136-8.  Google Scholar

[3]

E. Byrne, M. Greferath, A. Kohnert and V. Skachek, New bounds for codes over finite Frobenius rings,, Des. Codes Crypt., 57 (2010), 169.  doi: 10.1007/s10623-009-9359-y.  Google Scholar

[4]

E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings,, Des. Codes Crypt., 42 (2007), 289.  doi: 10.1007/s10623-006-9035-4.  Google Scholar

[5]

E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings,, in, (2010).   Google Scholar

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125.  doi: 10.1023/A:1008344232130.  Google Scholar

[7]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes,, IEEE Trans. Inform. Theory, 51 (2005), 2089.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[8]

I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers (in Russian),, Problemy Peredachi Informatsii, 33 (1997), 22.   Google Scholar

[9]

P. Delsarte, Weights of linear codes and strongly regular normed spaces,, Discrete Math., 3 (1972), 47.  doi: 10.1016/0012-365X(72)90024-6.  Google Scholar

[10]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.  doi: 10.1142/S0219498804000873.  Google Scholar

[11]

M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights,, Discrete Math., 289 (2004), 11.  doi: 10.1016/j.disc.2004.10.002.  Google Scholar

[12]

M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[13]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[14]

R. C. Heimiller, Phase shift pulse codes with good periodic correlation properties,, IRE Trans. Inform. Theory, IT-7 (1961), 254.  doi: 10.1109/TIT.1961.1057655.  Google Scholar

[15]

T. Honold, Characterization of finite Frobenius rings,, Arch. Math. (Basel), 76 (2001), 406.   Google Scholar

[16]

T. Honold, Further results on homogeneous two-weight codes,, in, (2007).   Google Scholar

[17]

T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999).   Google Scholar

[18]

B. R. McDonald, Finite rings with identity,, in, (1974).   Google Scholar

[19]

R. Raghavendran, Finite associative rings,, Compositio Math., 21 (1969), 195.   Google Scholar

[20]

J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions,, IEEE Trans. Inform. Theory, 52 (2006), 712.  doi: 10.1109/TIT.2005.862125.  Google Scholar

[1]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

[2]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[3]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049

[4]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124

[5]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[6]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[7]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[8]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[9]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046

[10]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003

[11]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[12]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

[13]

Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328

[14]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[15]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[16]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

[17]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129

[18]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[19]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[20]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]