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

2022-03-01
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.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES

# Suggestions

 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.