Optimal rate algebraic list decoding using narrow ray class fields

We use class field theory, specifically Drinfeld modules of rank 1, to construct a family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. Over a field of size ? 2 , these codes are within 2 / ( ? - 1 ) of the Singleton bound. The function fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are "folded" using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev Density Theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching 1 - R where R is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field.Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound - specifically, for any desired rate R ? ( 0 , 1 ) and constant e 0 , we get codes over an alphabet size ( 1 / e ) O ( 1 / e 2 ) that can be list decoded up to error fraction 1 - R - e confining close-by messages to a subspace with N O ( 1 / e 2 ) elements. Previous results for list decoding up to error-fraction 1 - R - e over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes themselves have the claimed list decoding property.

[1]  H. Stichtenoth,et al.  A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound , 1995 .

[2]  Venkatesan Guruswami,et al.  Folded codes from function field towers and improved optimal rate list decoding , 2012, STOC '12.

[3]  Venkatesan Guruswami,et al.  List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound , 2013, STOC '13.

[4]  Venkatesan Guruswami,et al.  Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes , 2013, IEEE Transactions on Information Theory.

[5]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[6]  W. W. Peterson,et al.  Encoding and error-correction procedures for the Bose-Chaudhuri codes , 1960, IRE Trans. Inf. Theory.

[7]  Venkatesan Guruswami Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate , 2010 .

[8]  Vijaya Kumar Murty,et al.  EFFECTIVE VERSIONS OF THE CHEBOTAREV DENSITY THEOREM FOR FUNCTION FIELDS , 1994 .

[9]  Claus Fieker,et al.  Computing equations of curves with many points , 2013 .

[10]  H. Niederreiter,et al.  Rational Points on Curves Over Finite Fields: Theory and Applications , 2001 .

[11]  H. Stichtenoth,et al.  On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields , 1996 .

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

[13]  Gerhard Frey,et al.  On the different of abelian extensions of global fields , 1992 .

[14]  Florian Hess,et al.  Computing Riemann-Roch Spaces in Algebraic Function Fields and Related Topics , 2002, J. Symb. Comput..