New identities relating wild Goppa codes

For a given support [email protected]?F"q"^"m^n and a polynomial [email protected]?F"q"^"m[x] with no roots in F"q"^"m, we prove equality between the q-ary Goppa codes @C"q(L,N(g))[email protected]"q(L,N(g)/g) where N(g) denotes the norm of g, that is g^q^^^m^^^-^^^1^+^...^+^q^+^1. In particular, for m=2, that is, for a quadratic extension, we get @C"q(L,g^q)[email protected]"q(L,g^q^+^1). If g has roots in F"q"^"m, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of g in F"q"^"m. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.

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

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  Pascal Véron.  Goppa Codes and Trace Operator , 1998, IEEE Trans. Inf. Theory.

[4]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

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

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

[7]  Sergey Bezzateev,et al.  A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension , 1998, Des. Codes Cryptogr..

[8]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[9]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[10]  Robert J. McEliece,et al.  Subspace subcodes of Reed-Solomon codes , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[11]  Tanja Lange,et al.  Wild McEliece Incognito , 2011, PQCrypto.

[12]  Pascal Véron.  Proof of Conjectures on the True Dimension of Some Binary Goppa Codes , 2005, Des. Codes Cryptogr..

[13]  Annett Baier.  Selected Areas in Cryptography , 2005, Lecture Notes in Computer Science.

[14]  Tanja Lange,et al.  Wild McEliece , 2010, IACR Cryptol. ePrint Arch..

[15]  Michael E. O'Sullivan,et al.  The dimension of subcode-subfields of shortened generalized Reed–Solomon codes , 2013, Des. Codes Cryptogr..

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

[17]  Henning Stichtenoth.  On the dimension of subfield subcodes , 1990, IEEE Trans. Inf. Theory.

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

[19]  Sergey Bezzateev,et al.  Chain of Separable Binary Goppa Codes and Their Minimal Distance , 2008, IEEE Transactions on Information Theory.

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

[21]  Sergey Bezzateev,et al.  Cumulative-Separable Codes , 2010, ArXiv.

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

[23]  Rudolf Lide,et al.  Finite fields , 1983 .

[24]  Qunying Liao.  On Reed-Solomon codes , 2011 .

[25]  Yves Edel,et al.  New code parameters from Reed-Solomon subfield codes , 1997, IEEE Trans. Inf. Theory.

[26]  Alain Couvreur,et al.  Codes and Cartier Operator , 2012, ArXiv.

[27]  Pascal Véron.  True Dimension of Some Binary Quadratic Trace Goppa Codes , 2001, Des. Codes Cryptogr..