 Generic
Number - 1528 |
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| Written
Date -
May 16th, 14 |
| Modified
Date -
May 16th, 14 |
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| Summary |
The paper introduces a new product of polynomials defined over a field. It is a
generalization of the ordinary product with inner polynomial getting non-overlapping
segments obtained by multiplying with coefficients and variable with expanding
powers. It has been called \textquoteleft{}Ordered Power Product\textquoteright{}
(OPP). Considering two rings of polynomials $R_{m}\left[x\right]=F\left[x\right]
$ modulo$x^{m}-1$ and $R_{n}\left[x\right]=F\left[x\right]$modulo$x^{n}-1$ ,
over a field F, the paper then considers the newly introduced product of the two
polynomial rings. Properties and algebraic structure of the product of two rings of
polynomials are studied and it is shown to be a ring. Using the new type of product
of polynomials, we define a new product of two cyclic codes and devise a method
of getting a cyclic code from the \textquoteleft{}ordered power
product\textquoteright{} of two cyclic codes. Conditions for the OPP of the
generators polynomials of component codes, giving a cyclic code are examined. It is
shown that OPP cyclic code so obtained is more efficient than the one that can be
obtained by Kronecker type of product of the same component codes.
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A New Class of Cyclic Codes using Ordered Power Product of Polynomials
