Parameters of Goppa codes revisited

We discuss parameters of Goppa (1970) codes, such as minimum distance, covering radius, distance distribution, and generalized Hamming weights. By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds on these parameters.

[1]  Elwyn R. Berlekamp,et al.  Goppa Codes , 2022 .

[2]  A. Tietäväinen,et al.  On the covering radius of long binary BCH codes , 1987 .

[3]  Jonathan I. Hall,et al.  The trace operator and redundancy of Goppa codes , 1992, IEEE Trans. Inf. Theory.

[4]  Marcel van der Vlugt,et al.  On generalized Hamming weights of BCH codes , 1994, IEEE Trans. Inf. Theory.

[5]  Charles T. Retter Bounds on Goppa codes (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[6]  Sergey Bezzateev,et al.  Subclass of binary Goppa codes with minimal distance equal to the design distance , 1995, IEEE Trans. Inf. Theory.

[7]  Simon Litsyn,et al.  On spectra of BCH codes , 1995, Eighteenth Convention of Electrical and Electronics Engineers in Israel.

[8]  Masao Kasahara,et al.  Further results on Goppa codes and their applications to constructing efficient binary codes , 1976, IEEE Trans. Inf. Theory.

[9]  Kenneth K. Tzeng,et al.  Characterization theorems for extending Goppa codes to cyclic codes (Corresp.) , 1979, IEEE Trans. Inf. Theory.

[10]  Kenneth K. Tzeng,et al.  On extending Goppa codes to cyclic codes (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[11]  Simon Litsyn,et al.  On the Covering Radius of Long Goppa Codes , 1995, AAECC.

[12]  Gérard D. Cohen,et al.  Covering Radius 1985–1994 , 1997, Applicable Algebra in Engineering, Communication and Computing.

[13]  G. Lachaud,et al.  The weights of the orthogonals of the extended quadratic binary Goppa codes , 1990, IEEE Trans. Inf. Theory.

[14]  MARCEL VAN DER VLUGT The true dimension of certain binary Goppa codes , 1990, IEEE Trans. Inf. Theory.

[15]  Michael Wirtz,et al.  On the parameters of Goppa codes , 1988, IEEE Trans. Inf. Theory.

[16]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[17]  J. H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[18]  G. David Forney,et al.  Dimension/length profiles and trellis complexity of linear block codes , 1994, IEEE Trans. Inf. Theory.

[19]  J. H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[20]  Charles T. Retter Intersecting Goppa codes , 1989, IEEE Trans. Inf. Theory.

[21]  Gérard D. Cohen,et al.  Upper bounds on generalized distances , 1994, IEEE Trans. Inf. Theory.

[22]  T. J. Rivlin Chebyshev polynomials : from approximation theory to algebra and number theory , 1990 .

[23]  Tor Helleseth,et al.  On the covering radius of cyclic linear codes and arithmetic codes , 1985, Discret. Appl. Math..

[24]  David M. Mandelbaum On the derivation of Goppa codes (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[25]  E. Berlekamp,et al.  Extended double-error-correcting binary Goppa codes are cyclic (Corresp.) , 1973, IEEE Trans. Inf. Theory.

[26]  Chin-Long Chen Equivalent irreducible Goppa codes (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[27]  Oscar Moreno,et al.  Exponential sums and Goppa codes: II , 1989, IEEE Trans. Inf. Theory.

[28]  Jean Conan,et al.  A transform approach to Goppa codes , 1987, IEEE Trans. Inf. Theory.

[29]  Oscar Moreno,et al.  Exponential sums and Goppa codes. I , 1991 .

[30]  A. Tietäinen On the covering radius of long binary BCH codes , 1987, Discret. Appl. Math..

[31]  Michael A. Tsfasman,et al.  Geometric approach to higher weights , 1995, IEEE Trans. Inf. Theory.