الفهرس 
Introduction and Basic ConceptsOnly 14 pages are availabe for public view
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Abstract
This thesis focuses on
• To study (θ−β)-constacyclic codes over the ring R = Fq +uFq +vFq +uvFq +v2Fq +uv2Fq , with u2 = 1, v3 = v, uv = vu, q = pm and p is an odd prime. The structural properties of (θ − β)-constacyclic codes over the ring R are studied. Further, generating polynomials and idempotent generators for (θ − β)-constacyclic codes over the ring R are studied, which appear in [Chapter 3, Section 3.4].
• To study quadratic residue codes over the ring R = Fp +uFp +vFp +uvFp +v2Fp +uv2Fp, with u2 = 1, v3 = v, uv = vu, and p is an odd prime.
The new results of chapter 3 are published in
Proceedings of the Jangjeon Mathematical Society (PJMS) Published By JANGJEON Mathematical Society
23 (2020), No. 1. pp. 71 - 79
Website http://jangjeonopen.or.kr/PJMS/open access.php
This thesis is divided into four chapters:
In chapter 1, we set some definitions which will be applied throughout the thesis, we study a class of rings called Frobenius rings and we define polynomial Rings and Skew polynomial Rings and give some basic theorem’s that will be needed through the thesis. Finally we define some famous codes that will appear through the thesis.
In chapter 2, we study linear and cyclic codes over Fq + uFq + vFq + uvFq , with u2 = u, v2 = v, uv = vu. Also give its Gray map and study linear and cyclic codes over F2 + uF2 + vF2 + uvF2 and F3 + uF3 + vF3 + uvF3, we study Constacyclic codes over Fq + uFq + vFq + uvFq , with u2 = u, v2 = v, uv = vu, and define a gray map over Fq + uFq + vFq + uvFq , and give their duals and self-dual constacylic codes over Fq + uFq + vFq + uvFq , we study Skew cyclic codes over Fq + uFq + vFq + uvFq where
u2 = u, v2 = v, uv = vu, q = pm and p is an odd prime, also study Skew cyclic codes
over Fq + uFq + vFq + uvFq , u2 = 1, v2 = 1, uv = vu, q = pm and p is an odd prime, and give the structural properties of skew cyclic codes over R through a decomposi- tion theorem. Finally, we study Skew (α1 + uα2 + vα3 + uvα4)-constacyclic codes over R = Fq + uFq + vFq + uvFq , where u2 = u, v2 = v, uv = vu, q = pm and p is an odd prime, and give some examples.
Chapter 3 focuses on study (θ − β)-constacyclic codes over the ring R = Fq + uFq + vFq + uvFq + v2Fq + uv2Fq , with u2 = 1, v3 = v, uv = vu, q = pm and p is an odd prime. The structural properties of (θ − β)-constacyclic codes over the ring R are studied. Fur- ther, generating polynomials and idempotent generators for (θ − β)-constacyclic codes over the ring R are studied.
Chapter 4 focuses on study quadratic residue codes over the ring R = Fp + uFp +
vFp + uvFp + v2Fp + uv2Fp, with u2 = 1, v3 = v, uv = vu, and p is an odd prime.
Glossary
Artinian ring: An R-module M is called artinian if it satisfies the descending chain condition (dcc) for submodules: any descending sequence M1 ⊃ M2 ⊃ . . . stabilizes (i.e., Mn = Mn+1 for sufficiently large n, or equivalently for some large n0 we have Mn = Mn0 for all n ≥ n0.
radical: rad(R) is the intersection of all the maximal left ideals of R.
socle: Given a left R-module M , its socle soc(M ) is the sum of all the simple submod- ules of M .
quasi-Frobenius: An Artinian ring R is quasi-Frobenius if there exists a permutation
σ of {1, 2, . . . , n} such that T (Rei) ∼= Soc(Reσ(i) and Soc(eiR) ∼= T (eσ(i)R).
character: Let G be a finite abelian group. In this article, a character is a group homo- morphism $ : G → Q/Z.
character group: The set of all characters of G.
dual code: The dual codeC⊥ of C is the set of n-tuples over R that are orthogonal to all codewords of C, i.e. C⊥ = {r|r.s = 0, ∀s ∈ C}.
self-dual code: The code C is called self-dual if C = C⊥.
submodule quotient: For any code of length n over R and any r ∈ R, we denote submodule quotient as follows: (C : r) = {s ∈ Rn|rs ∈ C}.
reciprocal polynomial: reciprocal polynomial of f denoted by f∗ is defined as f∗(x) =
xr+if(x−1) = arxi + ar
1xi+1 + . . . + a + ixr.
l-quasi-cyclic code: et σ be the cyclic shift on Rn defined by σ(r0, r1, . . . , rn
l
−1) =
(rn−1, r0, . . . , rn−2) and n =
n´l. A linear code which is invariant under σ
is called a
l-quasi-cyclic code of length n.
skew quasi-cyclic code: A code C length nm over Fq is said to be a skew quasi-cyclic
code of index m if πm(C) = C, where πm is the skew quasi-cyclic shift on (fn)m defined
as πm(a1| . . . |am) = (σ(a1)| . . . |σ(an)).
α−quasi-twisted code: Let C be a linear code of length n over R and n = ml, for m, l ∈
Z+ Then C is said to be an α−quasi-twisted code if for any (c0 0, c0 1, . . . , c0 l
−1, . . . , c
m−1 0
, cm−1 1, . . . , cm−1 l−1) ∈ C implies (αcm−1 0, αcm−1 1, . . . , αcm−1 l−1, . . . , cm−2 0, cm−2 1, . . .
, cm−1 l−2) ∈ C.
repetition code: A code C is the [n, 1] binary code consisting of two codewords 0 =
00 . . . 0 and 1 = 11 . . . 1.
quadratic residue codes or QR codes: special cases of duadic codes over Fq of odd prime length n = p, q must be a square modulo n.