210 research outputs found

    Codes and finite geometries

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    We explore the connections between finite geometry and algebraic coding theory, giving a rather full account of the Reed-Muller and generalized Reed-Muller codes. Some of the results and many of the proofs are new but this is largely an expository effort that relies heavily on the work of Delsarte et al. and of Charpin. The necessary geometric background is sketched before we begin the discussion of the Reed-Muller codes and their p-ary analogues. We prove all the classical results concerning these codes and include a discussion of the group-algebra approach and prove Berman's theorem characterizing the codes as powers of the radical. Included also is a discussion of the characterization of affine-invariant cyclic codes given by Kasami, Lin and Peterson and its generalization by Delsarte. our theme throughout this work is the relationship between these codes and the codes coming from both affine and projective geometries. The final section develops the theory in the more difficult case in which the field is not of prime order, here must look at subfield subcodes - which complicates the connection with the geometric codes, which are codesover the prime subfield of the field of the geometry

    Variations of the McEliece Cryptosystem

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    Two variations of the McEliece cryptosystem are presented. The first one is based on a relaxation of the column permutation in the classical McEliece scrambling process. This is done in such a way that the Hamming weight of the error, added in the encryption process, can be controlled so that efficient decryption remains possible. The second variation is based on the use of spatially coupled moderate-density parity-check codes as secret codes. These codes are known for their excellent error-correction performance and allow for a relatively low key size in the cryptosystem. For both variants the security with respect to known attacks is discussed

    New Quantum Codes from Evaluation and Matrix-Product Codes

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    Stabilizer codes obtained via CSS code construction and Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied, in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that can not be reached by them
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