By R. A. Hill

The purpose of this booklet is to supply an user-friendly therapy of the speculation of error-correcting codes, assuming not more than highschool arithmetic and the power to hold out matrix mathematics. The booklet is meant to function a self-contained path for moment or 3rd yr arithmetic undergraduates, or as a readable advent to the mathematical elements of coding for college students in engineering or computing device technology.

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This e-book – along side the volumes LNAI 8589 and LNBI 8590 – constitutes the refereed lawsuits of the tenth foreign convention on clever Computing, ICIC 2014, held in Taiyuan, China, in August 2014. The ninety two papers of this quantity have been conscientiously reviewed and chosen from various submissions.

**A First Course in Coding Theory**

The purpose of this e-book is to supply an hassle-free therapy of the idea of error-correcting codes, assuming not more than highschool arithmetic and the facility to hold out matrix mathematics. The e-book is meant to function a self-contained direction for moment or 3rd yr arithmetic undergraduates, or as a readable advent to the mathematical facets of coding for college students in engineering or desktop technological know-how.

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This result also holds for the more general class of CPSS formulas (deﬁnitions in the preliminaries section), showing that for this class of formulas isomorphism and counting have the same complexity. For the upper bound we use a recent result on the isometric dimension of partial cubes [18], the fact that GI is low for the class C= P [12], as well as the closure of this class under universal quantiﬁcation [10]. The hardness property uses a result of Curticapean [7], where it is proven that SameP M , the problem to decide if two given graphs have the same number of perfect matchings is complete for C= P.

For solution graphs of 2CNF formulas, isomorphism of a single connected component is exactly as hard as testing Graph Isomorphism. For a collection of such components (encoded by a single 2CNF formula), the isomorphism problem is complete for the complexity class C= P, a complexity class deﬁned in terms of exact counting. This means that deciding isomorphism of the solution graphs of 2CNF formulas is exactly as hard as testing if two such formulas have the same number of satisfying assignment. This result also holds for the more general class of CPSS formulas (deﬁnitions in the preliminaries section), showing that for this class of formulas isomorphism and counting have the same complexity.

2. (Mi ,j , Mi +1,j ) ∈ V for each (i , j ), (i + 1, j ) in S(m, n, i, j). Note that for simplicity we write Mi,j in place of M (i, j) to designate an entry of M . Intuitively, an (i, j)-partial (π, V, H)-solution for N is a function that colors the positions of N up to the entry (i, j) with elements from Σ in such a way that the vertical and horizontal constraints imposed by V and H respectively are respected. Mi,j2 be the string formed by all entries at the i-th row of M between positions j1 and j2 .