Bajalan, M.,
Martínez-moro, E.,
Sobhani, R.,
Szabo, S.,
Yılmazgüç, G.G. Discrete Mathematics (0012365X)347(1)
This paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the ⊥0-dual code for a given polycyclic code. © 2023 Elsevier B.V.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Garmsiri f.m., F.M.,
Khatami bidgoli, M.,
Sobhani, R. Communications in Algebra (00927872)51(3)pp. 1011-1019
A complex character χ of a finite group G is called sharp if (Formula presented.) where (Formula presented.). In this paper we give a characterization of finite groups all non-linear irreducible characters of which are sharp. © 2022 Taylor & Francis Group, LLC.
Abdollahi, A.,
Bagherian, J.,
Jafari, F.,
Khatami bidgoli, M.,
Parvaresh, F.,
Sobhani, R. Cryptography and Communications (19362447)15(5)pp. 891-903
We give two methods that are based on the representation theory of symmetric groups to study the largest size P(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on { 1 , ⋯ , n} with the minimum distance (at least) d under the Kendall τ -metric. The first method is an integer programming problem obtained from the transitive actions of Sn . The second method can be applied to refute the existence of perfect codes in Sn . Applying these methods, we reduce the known upper bound (n- 1) ! - 1 for P(n, 3) to (n-1)!-⌈n3⌉+2≤(n-1)!-2 , whenever n≥ 11 is prime. If n= 6 , 7, 11, 13, 14, 15, 17, the known upper bound for P(n, 3) is decreased by 3, 3, 9, 11, 1, 1, 4, respectively. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Communications in Algebra (00927872)50(6)pp. 2731-2739
Let χ be a virtual (generalized) character of a finite group G and (Formula presented.) be the image of χ on (Formula presented.) The pair (Formula presented.) is said to be sharp of type L or L-sharp if (Formula presented.) If the principal character of G is not an irreducible constituent of χ, the pair (Formula presented.) is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in 1988. This conjecture states that if (Formula presented.) is L-sharp and (Formula presented.) then the inner product (Formula presented.) is uniquely determined by L. We then prove that this conjecture is true in the case that (Formula presented.) is normalized, χ is a character of G, and L contains at least an irrational value. © 2022 Taylor & Francis Group, LLC.
Designs, Codes, and Cryptography (09251022)90(12)pp. 2841-2859
We study permutation codes which are groups and all of whose non-identity code elements have the same number of fixed points. It follows that over certain classes of groups such permutation codes exist. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Garmsiri f.m., F.M.,
Khatami bidgoli, M.,
Sobhani, R. Bulletin Of The Iranian Mathematical Society (10186301)48(6)pp. 3809-3821
For a finite group G and its character χ, let Lχ be the image of χ on G- { 1 }. The pair (G, χ) is said to be sharp of type L if | G| = Π a∈L(χ(1) - a) , where L= Lχ. The pair (G, χ) is said to be normalized if the principal character of G is not an irreducible constituent of χ. In this paper, we study normalized sharp pairs of type L= { - 1 , 1 , 3 } proposed by Cameron and Kiyota in [J Algebra 115(1):125–143, 1988], under some additional hypotheses. © 2022, The Author(s) under exclusive licence to Iranian Mathematical Society.
Discrete Mathematics (0012365X)345(4)
Let R be a finite commutative chain ring, D2n be the dihedral group of size 2n and R[D2n] be the dihedral group ring. In this paper, we completely characterize left ideals of R[D2n] (called left D2n-codes) when gcd(char(R),n)=1. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over R×S, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where R=Fp is a Galois field, we give a classification for left D2N-codes over Fp, for any positive integer N. In both cases we determine dual codes and identify self-dual ones. © 2021 Elsevier B.V.
In order to overcome the challenges posed by flash memories, the rank modulation scheme was proposed. In the rank modulation the codewords are permutations. In this paper, we study permutation codes with a specified length and minimum Kendall \tau-distance, and with as many codewords (permutations) as possible. We managed to make many significant improvements in the size of the best known codes. In particular, we show that for all n\geq 6 and for all \displaystyle \frac{3}{5}\begin{pmatrix}n\\2\end{pmatrix}\lt d\leq\frac{2}{3}\begin{pmatrix}n\\2\end{pmatrix} the largest size of a permutation code of length n and minimum distance at least d under Kendall \tau-metric is 4. © 2022 IEEE.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Khatami bidgoli, M.,
Shahbazi, Z.,
Sobhani, R. Czechoslovak Mathematical Journal (00114642)72(4)pp. 1081-1087
For a complex character χ of a finite group G, it is known that the product sh(χ)=∏l∈L(χ)(χ(1)−l) is a multiple of ∣G∣, where L(χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L(χ) and sh(χ) = ∣G∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. © 2022, Institute of Mathematics, Czech Academy of Sciences.
Designs, Codes, and Cryptography (09251022)87(10)pp. 2335-2340
Let M(n, d) be the maximum size of a permutation code of length n and distance d. In this note, the permutation codewords of a classical code C are considered. These are the codewords with all different entries in C. Using these codewords for Reed–Solomon codes, we present some good permutation codes in this class of codes. As a consequence, since these codes are subsets of Reed–Solomon codes, decoding algorithms known for Reed–Solomon codes can also be used as a decoding algorithm for them. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Cryptography and Communications (19362447)10(3)pp. 519-530
In this study, we consider the finite (not necessary commutative) chain ring R:=Fpm[u,θ]/u2, where θ is an automorphism of Fpm, and completely explore the structure of left and right cyclic codes of any length N over R, that is, left and right ideals of the ring S: = R[ x] / ;xN− 1. For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over R. Finally, we study Gray images of cyclic codes over R and as some examples, three linear codes over F4 with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over R. © 2017, Springer Science+Business Media, LLC.
Discrete Mathematics (0012365X)341(11)pp. 3106-3122
Let R be the finite chain ring R=Fp[u]∕〈u4〉 where p is a prime and m is a positive integer. In this work, we give a complete classification of (1+αu2)-constacyclic codes of length pk over R, where α is a nonzero element of Fp. We also completely determine self-dual such codes and enumerate them. Finally we discuss on Gray-maps on R which preserve self-duality, and also discuss on the images of self-dual constacyclic codes under these Gray maps. © 2018 Elsevier B.V.
Finite Fields and their Applications (10715797)39pp. 216-232
A matrix-product structure for repeated-root cyclic codes over finite fields is explored. Using this, some properties such as minimum distance and duality for these codes are rediscovered. Finally, a decoding algorithm is presented for this class of codes and the algorithm is modified and adapted for the binary repeated-root cyclic codes of length 2k. © 2016 Elsevier Inc.
Finite Fields and their Applications (10715797)34pp. 123-138
Let R be the finite chain ring Fpm[u]/〈u3〉, where p is a prime and m is a positive integer. In this study we completely determine the structure of (δ+αu2)-constacyclic codes of length pk over R, that is, ideals of the ring R[x]/〈xpk-(δ+αu2)〉, where δ and α are nonzero elements in Fpm. We show that when p is odd, there is no self-dual (δ+αu2)-constacyclic code of length pk over R and also in the case where p=2, self-dual codes exist when δ=1. We completely determine self-dual (1+αu2)-constacyclic codes of length 2k over F2m[u]/〈u3〉 and enumerate them. © 2015 Elsevier Inc. All rights reserved.
Matematika (01278274)(2)pp. 117-121
In 2001, Rosenblatt and Willis defined the concept of configuration of a group to give a condition for amenability of groups. In this paper, we study the relation between configuration and commutator subgroup G' of G and prove that if G(1) and G(2) are two finitely generated groups with the same configuration set, then cl c2 and if G'1 and G(2) are finite, then G'(1) G. Also, we prove that if two free finitely generated Burnside groups of finite exponent have the same configuration set, then they must be isomorphic.
IET Communications (17518628)8(12)pp. 2121-2130
In this study, using Group Permutation Low-Density Parity-Check (GP-LDPC) codes, the authors generalise the concept of array Low-Density Parity-Check (LDPC) codes from fields of prime order to those of prime power order. In fact, they consider the additive group of the finite field GF(q), q a prime power, as the underlying group for the GP-LDPC code construction and since when q is a prime, the author's code construction method coincides with that of quasi-cyclic array LDPC codes, they call their codes, generalised array LDPC (GA-LDPC) codes. First, they prove that, like array LDPC codes, GA-LDPC codes are quasi-cyclic codes. Then, they analyse the girth of GA-LDPC codes in a way similar to that for array LDPC codes and introduce some shortened GA-LDPC codes with girths 8, 10 and 12. For many values of g, J and L, the lengths of (J, L)-regular shortened GA-LDPC codes of girth g and rate at least 1 - J/L, constructed in this study, are smaller than the lengths of (J, L)-regular LDPC codes of girth g and rate at least 1 - J/L, constructed in the literature. Also, simulation results show that GA-LDPC codes perform well with the iterative message-passing decoding.
Let n be any positive integer and F-n be the friendship (or Dutch windmill) graph with 2n+1 vertices and 3n edges. Here we study graphs with the same adjacency spectrum as F-n. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let G be a graph cospectral with F-n. Here we prove that if G has no cycle of length 4 or 5, then G congruent to F-n. Moreover if G is connected and planar then G congruent to F-n. All but one of connected components of G are isomorphic to K-2. The complement (F-n) over bar, of the friendship graph is determined by its adjacency eigenvalues, that is, if (F-n) over bar is cospectral with a graph H, then H congruent to(F-n) over bar.
Turkish Journal of Mathematics (13036149)37(6)pp. 1061-1074
Let Rk,m be the ring F2m[u1,u2, . . . ,uk/〈u2i,uiUj - uiUj〉. In this paper, cyclic codes of arbitrary length n over the ring R2>m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m -basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n, n odd, over the ring Rk-i,m and cyclic codes of length n over the ring Rk,m. By determining the complete structure of cyclic codes of length 2 over R2,m , a mass formula for the number of these codes is given. Using this and the mentioned correspondence, the number of ideals of the rings R2,m and R3,m is determined. As a corollary, the number of cyclic codes of odd length n over the rings R2,m and R3,m is obtained. © Tübi̇tak.
IET Communications (17518628)6(12)pp. 1750-1756
In this study, a new method for constructing low-density parity-check (LDPC) codes is presented. This construction is based on permutation matrices which come from a finite abstract group and hence the codes constructed in this manner are called group permutation low-density parity-check (GP-LDPC) codes. A necessary and sufficient condition under which a GP-LDPC code has a cycle is given and some properties of these codes are investigated. A class of flexible-rate GP-LDPC codes without cycles of length four is also introduced. Simulation results show that GP-LDPC codes perform very well with the iterative decoding and can outperform their random-like counterparts. © 2012 The Institution of Engineering and Technology.
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences (17451337)93(4)pp. 808-813
A generalized Gray map for codes over the ring Fq〈u〉/ 〈ut+1〉 is introduced, where q = pm is a prime power. It is shown that the generalized Gray image of a linear length-N (1 - ut)-cyclic code over Fq[u]/〈ut+1〉 is a distance-invariant linear length-qtN quasi-cyclic code of index qt/p over Fq. It turns out that if (N, p) = 1 then every linear code over Fq that is the generalized Gray image of a length-N cyclic code over Fq[u]/〈ut+1〉, is also equivalent to a linear length-qtN quasi-cyclic code of index q t/p over Fq. The relationship between linear length-p N cyclic codes with (N, p) = 1 over Fp and linear length-N cyclic codes over Fp + uFp is explicitly determined. Copyright © 2010 The Institute of Electronics, Information and Communication Engineers.
Finite Fields and their Applications (10715797)15(3)pp. 387-391
Linear cyclic codes of length pk over the Galois ring GR (p2, m), that is ideals of the ring GR (p2, m) [u] / 〈 upk - 1 〉, are studied. The form of the dual codes is analyzed and self-dual codes are identified. © 2009 Elsevier Inc. All rights reserved.
Discrete Applied Mathematics (0166218X)157(13)pp. 2892-2903
This paper deals with cyclic codes over the Galois ring GR (p2, m). A unique set of generators for these codes and an algorithm for finding these generators are presented. The form of dual codes is studied. The obtained results on cyclic codes are extended to the class of negacyclic codes. © 2009 Elsevier B.V. All rights reserved.
Journal of Pure and Applied Algebra (00224049)212(4)pp. 727-734
In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which | G′ | is prime and G′ ≤ Z (G) as well as for groups G which | G′ | is prime and G′ ∩ Z (G) = 1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247]. © 2007 Elsevier Ltd. All rights reserved.
