On Codes based on BCK-algebras

Soft Computing, 2017

In the last time some papers were devoted to the study of the connections between binary block codes and BCK-algebras. In this paper, we try to generalize these results to n-ary block codes, providing an algorithm which allows us to construct a BCK-algebra from a given n-ary block code.

Information Sciences, 2011

The notion of a BCK-valued function on a set is introduced, and related properties are investigated. Codes generated by BCK-valued functions are established.

Soft Computing, 2020

In this paper we presented some connections between BCK-commutative bounded algebras, MV-algebras, Wajsberg algebras and binary block codes. Using connections between these three algebras, we will associate to each of them a binary block code and, in some circumstances, we will prove that the converse is also true.

Algorithmica, 2009

For a matrix *-algebra B, consider the matrix *-algebra A consisting of the symmetric tensors in the n-fold tensor product of B. Examples of such algebras in coding theory include the Bose-Mesner algebra and Terwilliger algebra of the (non)binary Hamming cube, and algebras arising in SDP-hierarchies for coding bounds using moment matrices. We give a computationally efficient block diagonalization of A in terms of a given block diagonalization of B, and work out some examples, including the Terwilliger algebra of the binary- and nonbinary Hamming cube. As a tool we use some basic facts about representations of the symmetric group.

In this paper, we will show some properties of codes over the ring $B_k = F_p[v_1,..., v_k ]/(v^2_ i = v_i, ∀i = 1,..., k$). These rings, form a family of commutative algebras over finite field $F_p$. We first discuss about the form of maximal ideals and characterization of automorphisms for the ring $B_k$. Then, we define certain Gray map which can be used to give a connection between codes over B k and codes over $F_p$. Using the previous connection, we give a characterization for equivalence of codes over $B_k$ and Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual codes over $B_k$ through MacWilliams relation of Hamming weight enumerator for such codes.

2018

Linear Algebra and its Applications, 1999

BCfi wdes over arbitrary finite ~~~nirnut:!tiv~ rings with identity arc drip& in LCF~S of their locator vector, The derivation is hased on the factorization of .I-' --I over the unit ring of an ~tppropr~lt~ extension of the finite rin g. We prcscnt an ~~~ci~nt,d~~iu~ procedure, based on the modified Berlekamp Massey ;li~~~rithrn. for that codes. The code construction and the decoding proccdurcs arc very similar to the BCH codes over finite integer rings. 43 1999 Ekxvier Scicncc Inc. Ail rights rwrwd. t f MS ~icl.Ev~~~~t~~it, 94BM: 94?35 Linear codes over rings have recently r&xd a great interest for their new role in algebraic coding theory and for their successful application in combined *Corr~s~ndin~ author. E-m& andr~detrt:mat.ithiI~~.unusy.hr, '

Communications of the Korean Mathematical Society, 2016

We define the notion of a residuated lattice valued function on a set as Jun and Song have done in BCK-algebras. We also investigate related properties of residuated lattice valued function. We establish the codes generated by residuated lattice valued function and conversely we give residuated lattice valued function and residuated lattice obtained by the giving binary block-code.

Anais de XXX Simpósio Brasileiro de Telecomunicações, 2012

For a non negative integer t, let A0 ⊂ A1 ⊂ • • • ⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of suitable Galois rings with multiplicative group A * i of units, and K0 ⊂ K1 ⊂ • • • ⊂ Kt−1 ⊂ Kt be the corresponding chain of unitary commutative rings, where each Ki is constructed by the direct product of corresponding residue fields of given Galois rings, with multiplicative groups K * i of units. This correspondence presents four different type of construction techniques of generator polynomials of sequences of BCH codes having entries from A * i and K * i for each i, where 0 ≤ i ≤ t. The BCH codes constructed in [1] are limited to given code rate and error correction capability, however, proposed work offers a choice for picking a suitable BCH code concerning code rate and error correction capability.

2021

In this paper, we are interested in the study of the right polycyclic codes as invariant subspaces of F n q by a fixed operator T R. This approach has helped in one hand to connect them to the ideals of the polynomials ring F q [x]/ f (X) , where f (x) is the minimal polynomial of T R. On the other hand, it allows to prove that the dual of a right polycyclic code is invariant by the adjoint operator of T R. Hence, when T R is normal we prove that the dual code of a right polycyclic code is also a right polycyclic code. However, when T R isn't normal the dual code is equivalent to a right polycyclic code. Finally, as in the cyclic case, the BCH-like and Hartmann-Tzeng-like bounds for the right polycyclic codes on Hamming distance are derived.