Browsing by Author "Muthoka, Geoffrey"

Now showing 1 - 9 of 9
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    Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets
    (UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, Jacob
    Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 7,8,9) acting on ordered 4-element subsets and Sn (n = 8,9,10) acting on ordered 5-element subsets.
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    Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets
    (UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, Jacob
    Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group 𝑆𝑛(𝑛 = 7,8,9) acting on ordered 4-element subsets and 𝑆𝑛 (𝑛 = 8,9,10) acting on ordered 5-element subsets .
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    Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples
    (UoEm, 2014) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein
    Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 5,6,7) and Sn (n = 6,7,8) acting on ordered pairs and triples respectively.
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    Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples
    (Uoem, 2014) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Muthoka, Geoffrey
    Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group 𝑆𝑛(𝑛 = 5,6,7) and 𝑆𝑛 (𝑛 = 6,7,8) acting on ordered pairs and triples respectively.
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    Cycle Index Formulas for D n Acting on Unordered Pairs
    (Uoem, 2015) Muthoka, Geoffrey; Kamuti, Ireri; Lao, Hussein; Kimani, Patrick Mwangi
    The cycle index of dihedral group D n acting on the set of X the vertices of a regular n- gon was studied (See [1]). In this paper we study the cycle index formulas of D n acting on unordered pairs from the set . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of and an odd value of .1.
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    Cycle Index Formulas for Dn Acting on Ordered pairs
    (UoEm, 2016-04) Muthoka, Geoffrey; Kamuti, Ireri; Kimani, Patrick; Hussein, Lao
    The cycle index of dihedral group Dn acting on the set of the vertices of a regular n-gon was studied by Harary and Palmer in 1973 [1]. Since then a number of researchers have studied the cycle indices dihedral group acting on different sets X={1,2,...,n} and the resulting formulas Dn have found applications in enumeration of a number of items. Muthoka (2015) [2] studied the cycle index formula of the –the n vertices of a regular -gon. In this paper we study the dihedral group acting on unordered pairs from the set X={1,2,..,n} cycle index formulas of acting on ordered pairs from the set . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of and an odd value of n .
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    Cycle Index Formulas for Dn Acting on Unordered Pairs
    (UoEm, 2015) Muthoka, Geoffrey; Kamuti, Ireri; Lao, Hussein; kimani, patrick
    The cycle index of dihedral group Dn acting on the set X of the vertices of a regular -gon was studied (See [1]). In this paper we study the cycle index formulas of Dn acting on unordered pairs from the set x . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of n and an odd value of n .
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    Enumeration of cyclic codes over GF(17)
    (UoEm, 2015-05) Hussein, Lao; Kivunge, Benard; Muthoka, Geoffrey; Mwangi, Patrick
    In this paper we seek the number of irreducible polynomials of xn− 1 over GF(17). We factorize Xn− 1 over GF(17)into irreducible polynomials using cyclotomic cosets of 17 modulo n . The number of irreducible polynomials factors of Xn− 1 over fq is equal to the number of q cyclotomic cosets of modulo n. Each monic divisor of Xn− 1 is a generator polynomial of cyclic code in Fqn. We show that the number of cyclic codes of length n over a finite field f is equal to Xn− 1. Lastly, the number of cyclic codes of length n , when n= 17 , = the number of polynomials that divide 17k ,n = 17k,n=17k − 1, ( , 17) = 1 are enumerated.
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    On the Number of Cyclotomic Cosets and Cyclic Codes over Z13
    (UoEm, 2018-06) Hussein, Lao; Kivunge, Benard; Kimani, Patrick; Muthoka, Geoffrey
    Let Zq be a finite field with q element and x n − 1 be a given cyclotomic polynomial. The number of cyclotomic cosets and cyclic codes has not been done in general. Although for different values of q the polynomial x n − 1 has been characterised. This paper will determine the number of irreducible monic polynomials and cyclotomic cosets of x n − 1 over Z13 .The factorization of x n − 1 over Z13 into irreducible polynomials using cyclotomic cosets of 13 modulo n will be established. The number of irreducible polynomials factors of x n − 1 over Zq is equal to the number of cyclotomic cosets of q modulo n. Each monic divisor of x n − 1 is a generator polynomial of cyclic code in Fq n . This paper will further show that the number of cyclic codes of length n over a finite field F is equal to the number of polynomials that divide x n − 1. Finally, the number of cyclic codes of length n, when n = 13k, n = 13k , n = 13k − 1, k, 13 = 1 are determine.