OOn Codes based on
BC K -algebras
A. Borumand Saeid, H. Fatemidokht, C. Flaut and M. Kuchaki Rafsanjani
Abstract.
In this paper, we present some new connections between
BCK -algebras and binary block codes.
Keywords:
BCI/BCK -algebras; Binary block codes; Partially ordered set.
AMS Classification.
BCI/BCK -algebras were first introduced in mathematics in 1966 by Y. Imaiand K. Iseki, through the paper [Im, Is; 66], as a generalization of the concept ofset-theoretic difference and propositional calculi. One of the recent applicationsof
BCK -algebras was given in the Coding Theory (see [Fl; 14], [Ju, So; 11]).
Definition 2.1.
An algebra ( X, ∗ , θ ) of type (2 ,
0) is called a
BCI -algebra ifthe following conditions are fulfilled: • BCI -1 (( x ∗ y ) ∗ ( x ∗ z )) ∗ ( z ∗ y ) = θ • BCI -2 ( x ∗ ( x ∗ y )) ∗ y = θ • BCI -3 x ∗ x = θ • BCI -4 x ∗ y = θ and y ∗ x = θ imply x = y a r X i v : . [ c s . I T ] D ec f a BCI -algebra X satisfies the following identity: • BCK -5 θ ∗ x = θ then X is called a BCK -algebra [Me, Ju; 94].The partial order relation on a
BCI / BCK -algebra is defined such that x ≤ y if and only if x ∗ y = θ. A BCI / BCK -algebra X is called commutative if x ∗ ( x ∗ y ) = y ∗ ( y ∗ x ) , for all x, y ∈ X and implicative if x ∗ ( y ∗ x ) = x, for all x, y ∈ X. If ( X, ∗ , θ ) and ( Y, ◦ , θ ) are two BCI / BCK -algebras, a map f : X → Y with the property f ( x ∗ y ) = f ( x ) ◦ f ( y ) , for all x, y ∈ X, is called a BCI / BCK -algebras morphism . If f is a bijective map, then f is an isomor-phism of BCI / BCK -algebras [Me, Ju; 94].Hereafter in this paper, X always denotes a finite BCI / BCK -algebra.In the following, we will use some notations and results given in the paper[Ju, So; 11].
Definition 2.2.
A mapping ˜ A : A → X is called a BCK -function on A, whichA and X is a nonempty set and a
BCK -algebra, respectively.
Definition 2.3.
A cut function of ˜ A , for q ∈ X , is defined to be a mapping˜ A q : A → { , } such that( ∀ x ∈ A )( ˜ A q ( x ) = 1 ⇔ q ∗ ˜ A ( x ) = θ ) Definition 2.4.
Let A = { , , . . . , n } and let X be a BCK -algebra. In [Ju, So;11], to each
BCK -function ˜ A : A → X can be associated a binary block-codeof length n . A codeword in a binary block-code V is v x = x x . . . x n such that x i = x j ⇔ A x ( i ) = j for i ∈ A and j ∈ { , } .Let v x = x x . . . x n and v y = y y . . . y n be two codewords belonging toa binary block-code V. Define an order relation (cid:54) c on the set of codewordsbelonging to a binary block-code V as follows [Ju, So; 11]: v x (cid:54) c v y ⇔ y i (cid:54) x i for i = 1 , , . . . , n. Main results
Definition 3.1.
Let ( S, (cid:54) ) be a partially ordered set. For q ∈ S , we define amapping S q : S → { , } such that( ∀ b ∈ S )( S q ( b ) = 1 ⇔ q (cid:54) b ) . A codeword v x = x x · · · x n of a binary block-code V is determined asfollow: x i = x j ⇔ S x ( i ) = j , for i ∈ S and j ∈ { , } . Example 3.2.
Let S = { , , , , } be a set with a partial order over S showedin the Figure 1(a). Figure 1: a)partial ordering. b)order relation (cid:54) c hen S s S S S S S V − P = { , , , , } . Example 3.3.
Let S = { , , , , } be a set with a partial order over S showedin the figure 2(a). Figure 2: a)partial ordering. b)order relation (cid:54) c hen S s S S S S S V − P = { , , , , } . Example 3.4.
Let S = { A, B, C, D } be a set with a partial order over S as inthe Figure 3(a). Figure 3: a)partial ordering. b)order relation (cid:54) c then S s A B C D S A S B S C S D V − P = { , , , } . In the following, we will compute binary block-code based on Definition 2.4.for
BCK -algebras. We will show that there is a correspondence between theordered relation on
BCK -algebra and partial ordered set.5 xample 3.5.
Let X = { , , , , } be a BCK -algebra with the following Cay-ley table: ∗ Figure 4: a)ordered relation. b)order relation (cid:54) c The above figure is the ordered relation on X.Let ˜ A : X → X be a BCK -function on X given by ˜ A = (cid:32) (cid:33) then ˜ A x ˜ A ˜ A ˜ A ˜ A ˜ A hus V − B = { , , , , } . Example 3.6.
Let X = { , , , , } be a BCK -algebra with the following Cay-ley table: ∗ Figure 5: a)ordered relation. b)order relation (cid:54) c The above figure is the ordered relation on X.Let ˜ A : X → X be a BCK -function on X given by ˜ A = (cid:32) (cid:33) then A x ˜ A ˜ A ˜ A ˜ A ˜ A V − B = { , , , , } . Remark 3.7.
On a partial ordered set with a minimum element θ we candefine a BCK -algebra structure(see [Fl; 14], (2 . V − P = V − B and V − P = V − B . We think that the problem occurred because we use onlythe order of BCK -algebra, not its algebraic properties. From above examples,it is obvious that the method presented in paper [Ju, So; 11] dose not dependon algebraic properties of
BCK -algebra. Also the obtained codes are not goodcodes, since their Hamming distance is not good. According to the figures 1to 5, there is a one-to-one correspondence between the ordering relation (cid:54) andorder relation (cid:54) c . Let X be a BCK -algebra and V be a linear binary block-code with n codewords of length n. We consider the matrix M V = ( m i,j ) i,j ∈{ , ,...,n } ∈M n ( { , } ) with the rows consisting of the codewords of V. This matrix is called the matrix associated to the code V. We consider the codewords in V lexico-graphic ordered in the ascending sense. With this remark, for V = { w , ....w n } , we denote lines in M V with L w , ..., L w n . Obviously, w = 00 ... (cid:124) (cid:123)(cid:122) (cid:125) n − time . On V, we de-fine the following multiplication ” ∗ ” w i ∗ w j = w k if and only if L w i + L w j = L w k . (2.1.) Proposition 3.8.
With this multiplication, ( V, ∗ , θ ) , where θ = w , becomes anabelian group. (cid:50) Remark 3.9.
The above group is a
BCI -algebra.8 xample 3.10.
We consider the binary linear code C = { , , , } = { θ, A, B, C } . The associated
BCI -algebra(group) is X = { θ, A, B, C } with zeroelement θ and multiplication given in the following table: ∗ θ A B Cθ θ A B CA A θ C BB B C θ AC C B A θ Definition 3.11.
Let ( X, ∗ , θ ) be a BCI / BCK -algebra, and I ⊆ X. We saythat I is a right-ideal if θ ∈ I and x ∈ I, y ∈ X imply x ∗ y ∈ I . An ideal I ofa BCI / BCK -algebra X is called a closed ideal if it is also a subalgebra of X (i.e. θ ∈ I and if x, y ∈ I it results that x ∗ y ∈ I ).Let C be a binary block code. In Theorem 2.9, from [Fl; 14], we find a BCK -algebra X such that the obtained binary block-code V X contains thebinary block-code C as a subset.Let C be a binary block code with m codewords of length q. With the abovenotations, let X be the associated BCK -algebra and W = { θ, w , ..., w m + q } theassociated binary block code which include the code C. We consider the code-words θ, w , w , ..., w m + q lexicographic ordered, θ ≥ lex w ≥ lex w ≥ lex ... ≥ lex w m + q . Let M ∈ M m + q +1 ( { , } ) be the associated matrix with the rows θ, w , ..., w m + q , in this order. We denote with L w i and C w j the lines and columns in the matrix M . The sub-matrix M (cid:48) of the matrix M with the rows L w , ..., L w m and thecolumns C w m +1 , ..., C w m + q − is the matrix associated to the code C. Proposition 3.12.
With the above notations, we have that { θ, w m +1 , ..., w m + q } determines a closed right ideal in the algebra X. Proof.
Let Y = { θ, w m +1 , ..., w m + q } . Due to the multiplications and the orderrelation (cid:22) given by the relations (2 .
1) and (1 .
1) from [Fl; 14], we can have onlythe following two possibilities: w i ∗ w j = θ or w i ∗ w j = w i . Therefore Y is a9ight-ideal in X. The multiplication (2 . . ) is : θ ∗ x = θ and x ∗ x = θ, ∀ x ∈ X ; x ∗ y = θ, if x ≤ y, x, y ∈ X ; x ∗ y = x, otherwise. (cid:50) Remark 3.13.
From Proposition 3.12, we obtain that to each binary blockcode we can associate a
BCK -algebra in which this code determines a rightideal.Let A be a nonempty set and X be a BCK -algebra.
Proposition 3.14.
Let C be a binary block code with m codewords of length q and let X be the associated BCK -algebra, as the above. Therefore, there arethe sets A and B ⊆ X, the BCK -function f : A → X and a cut function f r such that C = { f r : A → { , } / f r ( x ) = 1 , if and only if r ∗ f ( x ) = θ, ∀ x ∈ A, r ∈ B } . (cid:50) Remark 3.15. i) Let S = { , , ..., n } be the set with n elements. We knowthat ( P ( S ) , ∆ , ∩ ) is a Boolean ring, where P ( S ) is the power set of the set S, ∆ is symmetric difference of the sets and ∩ is the intersection of two sets.Let F = { f : S → { , } / f function } . To each f ∈ F corresponds a binaryblock codeword. To each binary block codeword c corresponds an elementfrom P ( S ). Indeed, to each binary codeword c = ( i , ..., i n ) we will associatethe set I c = { j , j , ..., j k } ∈ P ( S ) such that i j = i j = ... = i j k = 1 . ii) Using the above established correspondence, if C = { c , c , ..., c m } is alinear binary block code and Q = { I c , I c , ..., I c m } ⊆ P ( S ) , where I c i is theassociated subset for the codeword c , then Q is a sub-ring in the Boolean ring( P ( S ) , ∆ , ∩ ) . It results a bijective map between the sub-rings of the Booleanring ( P ( S ) , ∆ , ∩ ) and linear binary block codes with codewords of length n. Example 3.16. i) Let C = { , , , } = { w , w , w , w } be alinear binary block code and let X = { θ, w , w , w , w , w , w , w , w } be the btained BCK -algebra as in Theorem 2.9 from [Fl; 14]. The multiplication ofthis algebra is given in the below table ∗ θ w w w w w w w w θ θ θ θ θ θ θ θ θ θw w θ w w w w w θ θw w w θ w w w w θ w w w w w θ w w w w θw w w w w θ w w w w w w w w w w θ w w w w w w w w w w θ w w w w w w w w w w θ w w w w w w w w w w θ From Proposition 3.12, we remark that { θ, w , w , w , w } is a right idealin the BCK -algebra X. From Proposition 3.14, for A = { w , w , w , w } and B = { w , w , w , w } , we recover the initial code C. Example 3.17.
For the same linear binary block code C = { , , , } , let Q = { ∅ , { } , { } , { , }} as in Remark 3.15 ii). It is clear that Q is a sub-ring in the Boolean ring ( P ( { , , , } ) , ∆ , ∩ ) and C can be considered as asub-ring of this Boolean ring. Remark 3.18.
In [Fl; 14], Theorem 2.2, the studied binary block codes haveHamming distance equal with 1 . In the same paper, Theorem 2.9, to an arbitrarybinary block code C we associate a BCK algebra X and the code associated tothis algebra includes the code C. Proposition 3.14 improved this theorem sincewe can even obtain the code C and from Proposition 3.12 we have that the code C generate a right ideal in the algebra X. Remark 3.19.
The obtained results of above remarks and propositions canbe illustrated by partially ordered sets. Let C be a binary block code with m codewords of length q . According to Proposition 2.8 and Theorem 2.9 in [Fl;14], we can find the matrix M ∈ M m + q +1 ( { , } ) that is the matrix associatedto the code C . Let S be the associated partially ordered set. Therefore, thereare the sets A and B ⊆ S and the function f : A → S , such that we can definethe bellow set: C = { f r : A → { , } / f r ( b ) = 1 , if and only if r ≤ b, ∀ b ∈ A, r ∈ B } . A = { m + 2 , · · · , m + q + 1 } and B = { , · · · , m + 1 } . Example 3.20.
Let C = { , , , } be a linear binary block codeand let S = { , , , , , , , , } . In this example m = q = 4 . The matrixassociated to the code C is:1 1 1 1 1 1 1 1 10 1 0 0 0 0 0 1 10 0 1 0 0 0 0 1 00 0 0 1 0 0 0 0 10 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1 Figure 6: partial ordering.
The above figure is partial ordering over S. From above Proposition, A = { , , , } and B = { , , , } that from A and B, we can recover the initialcode C. onclusions. Even if, from the above examples, appears that the associatedbinary block codes depend only from the order relation defined on a
BCK -algebra, will be very interesting to study in a further paper how and if theproperties of
BCK -algebras can influence the properties of the associated binaryblock codes.
References [Im, Is; 66] Y. Imai, K. Iseki,
On axiom systems of propositional calculi ,Proc. Japan Academic, , 19-22.[Fl; 14] C. Flaut,
BCK -algebras arising from block codes, arxiv.[Ju, So; 11] Y. B. Jun, S. Z. Song,
Codes based on
BCK -algebras , Inform.Sciences., , 5102-5109.[Me, Ju; 94] J. Meng and Y.B. Jun,
BCK -algebras-algebras