Abstract
A function associated with a linear error-correcting code defined over a finite field with q elements, where q is a power of a prime p, is introduced. This function is a generalization of the energy function associated with a linear block code over the field ℤp as described by Bruck and Blaum in [1]. It is proven to have similar properties. In particular, the Maximum Likelihood Decoding (MLD) problem is shown to be characterized by the maximization of this function.
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Bruck, J., Blaum, M.: Neural Networks, Error-Correcting Codes, and Polynomials over the Binary n-Cube. IIIE Trans. on Inf. Theory, vol. 35, No.5 (1989) 976–987
Bruck, J., Sanz, J.: A study of neural networks. Int. J. Intelligent Systems, vol.3 (1988) 59–75
Bruck, J., Goodman, J.W.: A generalized convergence theorem for neural networks. IIIE Trans. Inf. Theory, vol.IT-34 (1988) 1089–1092
Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. (USA) vol.79 (1986) 269–281
Goppa, V.D.: Algebraico-Geometric codes. Math. URSS-Izv 21(1) (1983) 75–91
Goppa, V.D.: Geometry and Codes. Mathematics and Its Applications. Kluwer Academic Publishers, (1988)
Manin, Y.I., Vlädut, S.G.: Linear codes and modular curves. J.Sov. Math. 30 (6) (1985) 2611–2643
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York, (1977)
Moreno, C. Algebraic curves over finite fields. Cambridge U. Press, (1991)
Peterson, W.W. and Weldon, W., J. Error-Correcting Codes. Cambridge, MA.,MIT Press, (1971)
Rentería, C., Tapia-Recillas, H.: A lifting of BCH codes over a hyperelliptic curve. To appear in Marshall Hall Memorial Conference. (J. Wiley & Sons Co.)
Rentería, C., Tapia-Recillas, H.; A geometric code on an Artin-Schreier curve. C. Numerantium, vol.84, Dec.1991, 3–7, Winnipeg, Canada
Stichtenoth, H.: Self-dual Goppa codes. J. of Pure and Applied Algebra 55 (1988) 199–211
van Lint, J.H. Introduction to Coding Theory. GTM 86, Springer-Verlag, New York, (1982)
van Lint, J.H., van der Geer.: Introduction to Coding Theory and Algebraic Geometry Birkhäuser Verlag, (1988)
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© 1993 Springer-Verlag Berlin Heidelberg
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Rentería, C., Tapia-Reculas, H. (1993). Energy functions associated with error-correcting codes. In: Cohen, G., Mora, T., Moreno, O. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1993. Lecture Notes in Computer Science, vol 673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56686-4_49
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DOI: https://doi.org/10.1007/3-540-56686-4_49
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