Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes

Koese, Seyda
Özbudak, Ferruh
We determine the unique factorization of some polynomials over a finite local commutative ring with identity explicitly. This solves and generalizes the main conjecture of Qian, Shi and Sole in [13]. We also give some applications to enumeration of certain generalized double circulant self-dual and linear complementary dual (LCD) codes over some finite rings together with an application in asymptotic coding theory.


Factorization of Joint Probability Mass Functions into Parity Check Interactions
Bayramoglu, Muhammet Fatih; Yılmaz, Ali Özgür (2009-07-03)
We show that any joint probability mass function (PMF) can be expressed as a product of parity check factors an d factors of degree one with the help of some auxiliary variables, if the alphabet size is appropriate for defining a parity chec k equation. In other words, marginalization of a joint PMF is equivalent to a soft decoding task as long as a finite field can be constructed over the alphabet of the PMF. In factor graph terminology this claim means that a factor graph representing such a joint PMF alw...
On decoding interleaved reed-solomon codes
Yayla, Oğuz; Özbudak, Ferruh; Department of Cryptography (2011)
Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher-Kiayias-Yung is extended to the polynomials whose degrees are allowed to be distinct. Furthermore, it is observed that probability of the algorithm can be increased. Specifically, for a finite field $\F$, we present a probabilistic algorithm which can recover polynomials $p_1,\ldots, p_r \in \F[x]$ of degree less than $k_1,k_2,\ldots,k_r$, respectively with given field evaluations $p_l(z_i) = y_{i,l}$ for all $i \in I$, $
Additive polynomials and primitive roots over finite fields
Özbudak, Ferruh (2001-01-01)
We prove existence of primitive roots with a prescribed nonzero image using the arithmetic of algebraic function fields for a class of polynomials over sufficiently large finite fields.
Switchings of semifield multiplications
Hou, Xiang-dong; Özbudak, Ferruh; ZHOU, Yue (2016-08-01)
Let B(X, Y) be a polynomial over F-qn which defines an F-q-bilinear form on the vector space F-qn, and let xi be a nonzero element in F-qn. In this paper, we consider for which B(X, Y), the binary operation xy + B(x, y) xi defines a (pre)semifield multiplication on F-qn. We prove that this question is equivalent to finding q-linearized polynomials L(X) is an element of F-qn [X] such that Tr-qn/q (L(x)/x) not equal 0 for all x is an element of F-qn*. For n <= 4, we present several families of L(X) and we inv...
On the number of topologies on a finite set
Kızmaz, Muhammet Yasir (2019-01-01)
We denote the number of distinct topologies which can be defined on a set X with n elements by T(n). Similarly, T-0(n) denotes the number of distinct T-0 topologies on the set X. In the present paper, we prove that for any prime p, T(p(k)) k+ 1 (mod p), and that for each natural number n there exists a unique k such that T(p + n) k (mod p). We calculate k for n = 0, 1, 2, 3, 4. We give an alternative proof for a result of Z. I. Borevich to the effect that T-0(p + n) T-0(n + 1) (mod p).
Citation Formats
S. Koese and F. Özbudak, “Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes,” CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES, pp. 0–0, 2022, Accessed: 00, 2022. [Online]. Available: https://hdl.handle.net/11511/96945.