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Within the new period of expertise and complex communications, coding concept and cryptography play a very major position with an immense volume of analysis being performed in either parts. This ebook offers a few of that examine, authored through well-liked specialists within the box. The booklet includes articles from numerous issues such a lot of that are from coding conception. Such themes contain codes over order domain names, Groebner illustration of linear codes, Griesmer codes, optical orthogonal codes, lattices and theta features regarding codes, Goppa codes and Tschirnhausen modules, s-extremal codes, automorphisms of codes, and so on. There also are papers in cryptography which come with articles on extremal graph conception and its functions in cryptography, quickly mathematics on hyperelliptic curves through persevered fraction expansions, and so forth. Researchers operating in coding idea and cryptography will locate this booklet a very good resource of knowledge on contemporary study.

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2) If d is even then either there exists g ∈ G such that c = cg and Td (g) = t + 1 or there exist g1 , g2 ∈ G such that c = cg1 + cg2 = ψ(τ1 ) + ψ(τ2 ), where g1 = τ1 − τ , g2 = τ2 − τ (τ1 = T (g1 ), τ2 = T (g2 ), τ = Can(g1 , G) = Can(g2 , G)), with t + 1 = Td (g1 ) = Td (g2 ). A codeword c is called minimal if does not exist c1 ∈ C \ {c} such that supp(xc1 ) ⊂ supp(xc ). Then we have the following result for a set of Gr¨ obner codewords. 3. The elements of the set CG of Gr¨ obner codewords are minimal codewords of the code C.

In that paper several heuristic and incremental procedures are shown for dealing with the Code Equivalence Problem (some of them are implemented in the package GBLA-LC [6]). In the binary case we can make use of the reduced basis. The main idea is the following, if two codes are equivalent then, under the appropriate May 10, 2007 8:8 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 30 permutation, words of the same weight must be sent to each other. Note also, that it will be used only the level t + 1 of the reduced bases, which is the first interesting level, from level 1 to t all the elements are canonical forms (we define level l of a reduced basis as the set of binomials of the reduced basis which their maximal terms have cardinal of the set of indices equal to l).

1–41 (To appear, 2007). [6] M. Borges-Quintana, M. A. Borges-Trenard and E. Mart´ınez-Moro. GBLALC: Gr¨ obner basis by linear algebra and codes. International Congress of Mathematicians 2006 (Madrid), Mathematical Software, EMS (Ed), 604– 605, 2006. es/~edgar/GBLAweb/. C. Faug`ere, P. Gianni, D. Lazard, T. Mora. Efficient Computation of ZeroDimensional Grobner Bases by Change of Ordering. J. , vol. 16(4), 329–344, 1993. [8] The GAP Group. 9, 2006. org. [9] Y. Han. A New Decoding Algorithm for Complete Decoding of Linear Block Codes.