Abstract
This contribution gives an introduction to algebraic coding theory over rings. We will start with a historical sketch and then present basics on rings and modules. Particular attention will be paid to weight functions on these, before some foundational results of ring-linear coding will be discussed. Among these we will deal with code equivalence, and with MacWilliams’ identities about the relation between weight enumerators. A further section is devoted to existence bounds and code optimality. An outlook will then be presented on the still unsolved problem of the construction of large families of ring-linear codes of high quality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Aigner, Combinatorial theory, Springer, Berlin, 1997.
Y. Al-Khamees, The enumeration of finite chain rings, Panamer. Math. J. 5 (1995), no. 4, 75–81.
E. F. Assmus Jr. and H. F. Mattson, Error-correcting codes: An axiomatic approach, Information and Control 6 (1963), 315–330.
M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, Addison–Wesley, London, 1969.
D. Augot, E. Betti, and E. Orsini, An introduction to linear and cyclic codes, this volume, 2009, pp. 47–68.
I. F. Blake, Codes over certain rings, Information and Control 20 (1972), 396–404.
I. F. Blake, Codes over integer residue rings, Information and Control 29 (1975), 295–300.
A. Bonnecaze, P. Solé, C. Bachoc, and B. Mourrain, Type II codes over Z 4, IEEE Trans. on Inf. Th. 43 (1997), no. 3, 969–976.
A. Bonnecaze, E. Rains, and P. Solé, 3-colored 5-designs and Z 4 -codes, J. Statist. Plann. Inference 86 (2000), no. 2, 349–368.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.
B. Buchberger, Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional systems theory, Reidel, Dordrecht, 1985, pp. 184–232.
B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.
E. Byrne and P. Fitzpatrick, Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001), no. 5, 565–584.
E. Byrne and P. Fitzpatrick, Hamming metric decoding of alternant codes over Galois rings, IEEE Trans. on Inf. Th. 48 (2002), no. 3, 683–694.
E. Byrne and T. Mora, Gröbner bases over commutative rings and applications to coding theory, this volume, 2009, pp. 239–261.
E. Byrne, M. Greferath, and M. E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr. 42 (2007), no. 3, 289–301.
A. R. Calderbank and G. M. McGuire, Construction of a (64,237,12) code via Galois rings, Des. Codes Cryptogr. 10 (1997), no. 2, 157–165.
A. R. Calderbank and N. J. A. Sloane, Modular and p -adic cyclic codes, Des. Codes Cryptogr. 6 (1995), no. 1, 21–35.
C. Carlet, \({\mathbb{z}}_{2^{k}}\) -linear codes, IEEE Trans. on Inf. Th. 44 (1998), 1543–1547.
W. E. Clark and D. A. Drake, Finite chain rings, Abh. Math. Sem. Univ. Hamburg 39 (1973), 147–153.
W. E. Clark and J. J. Liang, Enumeration of finite commutative chain rings, J. Algebra 27 (1973), 445–453.
P. M. Cohn, Algebra I, Wiley, New York, 1989.
I. Constantinescu, Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik, Ph.D. thesis, Technische Universität München, 1995.
S. T. Dougherty, M. Harada, and P. Solé, Shadow codes over ℤ4, Finite Fields Appl. 7 (2001), no. 4, 507–529.
I. M. Duursma, M. Greferath, S. Litsy, and S. E. Schmidt, A \({\mathbb{z}}_{9}\) -linear code inducing a ternary (72,325,24)-code, Proc. of OC2001, 1999.
I. M. Duursma, M. Greferath, S. N. Litsyn, and S. E. Schmidt, A \({\mathbb{z}}\sb8\) -linear lift of the binary Golay code and a nonlinear binary (96,237,24)-code, IEEE Trans. on Inf. Th. 47 (2001), no. 4, 1596–1598.
M. Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), nos. 1–3, 273–277.
M. Greferath and M. E. O’Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math. 289 (2004), nos. 1–3, 11–24.
M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams’ equivalence theorem, J. Combin. Theory Ser. A 92 (2000), no. 1, 17–28.
M. Greferath, A. Nechaev, and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247–272.
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The Z 4 -linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. on Inf. Th. 40 (1994), no. 2, 301–319.
T. Helleseth and P. V. Kumar, The algebraic decoding of the Z 4 -linear Goethals codes, IEEE Trans. on Inf. Th. 41 (1995), no. 6, 2040–2048, part 2.
T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415.
P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo p m, Finite Fields Appl. 3 (1997), no. 4, 334–352.
A. M. Kerdock, A class of low-rate nonlinear binary codes, Information and Control 20 (1972a), 182–187.
A. M. Kerdock, Information and Control 21 (1972b), 395.
M. Klemm, Über die Identität von MacWilliams für die Gewichtsfunktion von Codes, Arch. Math. (Basel) 49 (1987), no. 5, 400–406.
W. Krull, Algebraische Theorie der Ringe II, Math. Ann B 91 (1923), 1–46.
T. Y. Lam, A first course in noncommutative rings, Springer, New York, 1991.
T. Y. Lam, Lectures on modules and rings, Springer, Berlin, 1999.
P. Langevin and P. Solé, Duadic Z 4 -codes, Finite Fields Appl. 6 (2000), no. 4, 309–326.
F. J. MacWilliams, Combinatorial properties of elementary Abelian groups, Ph.D. thesis, Radcliffe College, Cambridge MA, 1962.
F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, vols. I and II, North-Holland, Amsterdam, 1977.
E. Martínez-Moro and I. F. Rúa, Multivariable codes over finite chain rings: serial codes, SIAM J. Discrete Math. 20 (2006), no. 4, 947–959.
T. Muegge, Existenzschranken für Blockcodes über endlichen Frobeniusringen, Diplomarbeit, 2002.
A. A. Nechaev, Kerdock codes in cyclic form, Discrete Math. Appl. 1 (1991), no. 4, 365–384.
G. H. Norton and A. Sălăgean, Strong Gröbner bases and cyclic codes over a finite-chain ring, International Workshop on Coding and Cryptography (Paris, 2001), Electron. Notes Discrete Math., vol. 6, Elsevier, Amsterdam, 2001a, p. 11.
G. H. Norton and A. Sălăgean, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Austral. Math. Soc. 64 (2001b), no. 3, 505–528.
G. H. Norton and A. Sălăgean, Gröbner bases and products of coefficient rings, Bull. Austral. Math. Soc. 65 (2002), no. 1, 145–152.
G. H. Norton and A. Sălăgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl. 9 (2003), no. 2, 237–249.
F. P. Preparata, A class of optimum nonlinear double-error correcting codes, Information and Control 13 (1968), 466–473.
C. Rong, T. Helleseth, and J. Lahtonen, On algebraic decoding of the Z 4 -linear Calderbank-McGuire code, IEEE Trans. on Inf. Th. 45 (1999), no. 5, 1423–1434.
G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.
L. H. Rowen, Ring theory, Academic, San Diego, CA, 1991.
A. Sălăgean-Mandache, On the isometries between \(Z\sb{p\sp k}\) and Z k p , IEEE Trans. on Inf. Th. 45 (1999), no. 6, 2146–2148.
C. Satyanarayana, Lee metric codes over integer residue rings, IEEE Trans. on Inf. Th. 25 (1979), 250–253.
P. Shankar, On BCH codes over arbitrary integer rings, IEEE Trans. on Inf. Th. 25 (1979), no. 4, 480–483.
K. Shiromoto, Singleton bounds for codes over finite rings, J. Algebraic Combin. 12 (2000), no. 1, 95–99.
K. Shiromoto and L. Storme, A Griesmer bound for linear codes over finite quasi-Frobenius rings, Discrete Appl. Math. 128 (2003), no. 1, 263–274, WCC 2001.
E. Spiegel, Codes over ℤ m , Information and Control 35 (1977), 48–51.
E. Spiegel, Codes over ℤ m , revisited, Information and Control 37 (1978), 100–104.
R. P. Stanley, Enumerative combinatorics, vol. 1, Cambridge University Press, Cambridge, 1997, With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original.
J. A. Wood, Extension theorems for linear codes over finite rings, Applied algebra, algebraic algorithms and error-correcting codes (Toulouse, 1997), Springer, Berlin, 1997, pp. 329–340.
J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Greferath, M. (2009). An Introduction to Ring-Linear Coding Theory. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-93806-4_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93805-7
Online ISBN: 978-3-540-93806-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)