© Hindawi Publishing Corp. MINIMUM DISTANCES OF ERROR-CORRECTING CODES IN INCIDENCE RINGS

The main theorem of this paper gives a formula for 
the largest minimum distance of error-correcting codes 
considered as ideals in incidence rings defined by 
directed graphs.

It is very well known that additional algebraic structure can give advantages for coding applications.For example, all cyclic error-correcting codes are principal ideals in the group algebras of cyclic groups (see the survey [4] and the books [3,5,6,7]).Serious attention in the literature has been devoted to considering properties of ideals in various ring constructions essential from the point of view of coding theory.The aim of this paper is to obtain a formula for the largest minimum distance of ideals in incidence rings defined by directed graphs.
Let R be a ring with identity element 1, and let D = (V , E) be any graph with the set V = {1,...,n} of vertices and a set E ⊆ V × V of edges.We use the standard definition of an incidence ring (see, e.g., [3,Section 3.15]).The incidence ring I(D, R) is the free left R-module with basis consisting of all edges in E, where multiplication is defined by the distributive law and the rule for all x, y, x, t ∈ V .The graph D is said to be balanced if for all x 1 ,x 2 ,x 3 ,x 4 ∈ V with (x 1 ,x 2 ), (x 2 ,x 3 ), (x 3 ,x 4 ), (x 1 ,x 4 ) ∈ E, It is proved in [1] that I(D, R) is an associative ring if and only if D is balanced.
For any vertex v ∈ V , we introduce the following sets of vertices: Denote by E down the set of all edges (x, y) ∈ E such that there exists z ∈ V with (z, x), (z, y) ∈ E. Let E up be the set of all edges (x, y) ∈ E such that there exists z ∈ V with (x, z), (y, z) ∈ E.
For each vertex v ∈ V and a subset S ⊆ Out(v), denote by In S (v) the set of all x ∈ V such that the following conditions hold: (I2) for every y ∈ Out(v), (x, y) ∈ E if and only if y ∈ S. Similarly, for each S ⊆ In(v), denote by Out S (v) the set of all y ∈ V such that the following conditions hold: The minimum distance is worth considering from the viewpoint of coding theory, because it gives the number of errors a code can detect or correct.Denote by wt(x) the Hamming weight of an element x ∈ M n (F ), that is, the number of edges (u i ,v i ) with nonzero coefficients r i in the standard record x = n i=1 r i (u i ,v i ).The Hamming distance between two elements and the minimum distance of a code are then defined in the usual way.The distance between two elements is the Hamming weight of their difference.The minimum distance dist(C) of a code C is the minimum distance between a pair of distinct elements in the code.If a code is a linear space, then its minimum distance is equal to the minimum weight of a nonzero element in the code.An ideal is said to be principal if it is generated by one element.This property is also convenient since, in order to store the whole code in computer memory, it is enough to record only one generator.Besides, the generators of codes are used in encoding and decoding algorithms.This is why it is nice that the best minimum distances for all ideals in incidence rings are achieved by principal ideals, as the following main theorem shows.We do not assume that all vertices of the graph have loops since, otherwise, all ideals of the incidence ring have minimum distance one, and being regarded as codes they cannot detect even one error.

Theorem 1. Let D = (V , E) be a balanced graph, and let R be a ring with identity element. Then the incidence ring I(D, R) has a principal ideal with minimum distance
and the minimum distances of all ideals of I(R, D) do not exceed dist(D).
Proof.In the first part of the proof, we show that the incidence ring I(D, R) always contains a principal ideal with the minimum distance given by (4).
First, consider the ideal A generated in I(D, R) by the element a = (x,y)∈E 0 (x, y), where we assume that a = 0 if E 0 = ∅.If K is an associative ring not necessarily containing an identity element, then the left and right annihilators of K are the sets The annihilator of the ring K is the set defined by The definitions of E up , E down , E 0 , and (1) imply that the following inclusions hold: It follows from (9) that the ideal A is equal to the subring generated by a.
Hence, the minimum distance of A is equal to the weight of a, that is, (10) Indeed, each nonzero element x in the ideal B v,S can be written in the form where r ,r i ,r j ,r k ∈ R and x i ,y i ,z j ,w j ,u k ,v k ,e k ,f k ∈ V , for all i, j, k.The definition of b v,S , condition (I1) and inclusion (8 By the definition of b v,S and (1), we may remove all remaining zero summands and assume that z Fix each j ∈ {1,...,m} and every x ∈ In S (v), condition (I2) yields that We may assume that terms that differ only in a coefficient in R have been combined in (13).If the same edge occurs in the elements b v,S (v, w j ) and b v,S (v, w k ), then (1) implies that w j = w k , and so b v,S (v, w j ) = b v,S (v, w k ), a contradiction.Similarly, if b v,S and b v,S (v, w j ) have a common edge, then w j = v and b v,S = b v,S (v, w j ), a contradiction again.This establishes (10).Third, a similar argument shows that for each v ∈ V and S ⊆ Out(v), there exists an ideal with minimum distance | Out S (v)|.Indeed, to this end, it suffices to consider the ideal C v,S generated by c v,S = x∈Out S (v) (v, x), where we assume that c v,S = 0 if Out S (v) = 0.A verification analogous to the one carried out in the preceding case shows that the minimum distance of C v,S is given by Obviously, the principal ideal P generated by any element (x, y) ∈ E has minimum distance 1.If we choose an ideal with largest minimal distance among the principal ideals P , A, B v,S , and C v,S , then we get the distance in (4) equal to max 1, dist(A), max In the second part of the proof, we take an arbitrary ideal K of I(D, R) and show that the distance of K is less than or equal to the one given by (4).Choose a nonzero element w = n i=1 r i (x i ,y i ) with minimum weight in K, where 0 = r i ∈ R, (x i ,y i ) ∈ E, for i = 1,...,n.We have to verify that the weight n of w does not exceed the maximum in (4).Clearly, we may assume that dist(K) > 1, and so n > 1.
If (x i ,y i ) ∈ E 0 for all i = 1,...,n, then n ≤ |E 0 |, and we are done.Further, assume that at least one of the edges in the expansion of w, say (x 1 ,y 1 ), is not in E 0 .Then (x 1 ,y 1 ) belongs to E down ∪ E up .
Our main theorem indicates that dist(D) is the maximum of four values.Next, we are going to give small examples showing that it is impossible to remove any of these four values from the formula.The following four graphs D 1 , D 2 , D 3 , and D 4 are defined by their adjacency matrices A 1 , A 2 , A 3 , and A 4 , where The four values that occur in formula (4) for the graphs D 1 , D 2 , D 3 , and D 4 are equal to (1, 0, 0, 0), (1, 2, 0, 0), (1, 0, 2, 0), and (1, 0, 0, 2), respectively.Every incidence ring can be thought of as a contracted semigroup ring (see [3]).Let S be a finite semigroup.Recall that the semigroup ring F[S] consists of all sums of the form s∈S r s s, where r s ∈ F for all s ∈ S, with addition and multiplication defined by the rules If S is a semigroup with zero θ, then the contracted semigroup ring F 0 [S] is the quotient ring of F[S] modulo the ideal Fθ.Thus, F 0 [S] consists of all the sums of the form θ =s∈S r s s, and all the elements of Fθ are identified with zero.
A graph D = (V , E) defines an associative incidence ring if and only if the set forms a semigroup with respect to the operation defined by (1), and therefore both of these properties are equivalent to the graph being balanced.Further, it is easily seen that the incidence ring I(D, F) is isomorphic to the contracted semigroup ring F 0 [S D ].Thus, our paper also contributes to the investigation of coding properties of ideals in semigroup rings started in [2].

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.