Abstract:
Irreducible trinomials of given degree n over $F_2$ do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over $F_2$. A condition for divisibility of self-reciprocal trinomials by irreducible polynomials over $F_2$ is established. And we extend Welch's criterion for testing if an irreducible polynomial divides trinomials $x^m+x^s+1$ to the trinomials $x^{am}+x^{bs}+1$.

Abstract:
Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree , where , and exactly one of？？ ,？？and？？ is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2. 1. Introduction Irreducible polynomials of low weight over a finite field are frequently used in many applications such as coding theory and cryptography due to efficient arithmetic implementation in an extension field and, thus, it is important to determine the irreducibility of such polynomials. The weight of a polynomial means the number of its nonzero coefficients. Characterization of the parity of the number of irreducible factors of a given polynomial is of significance in this context. If a polynomial has an even number of irreducible factors, then it is reducible and, thus, the study on the parity of this number can give a necessary condition for irreducibility. Swan [1] gives the first result determining the parity of the number of irreducible factors of trinomials over . Vishne [2] extends Swan’s theorem to trinomials over an even-dimensional extension of . Many Swan-like results focus on determining the reducibility of higher weight polynomials over ; see for example [3, 4]. Some researchers obtain the results on the reducibility of polynomials over a finite field of odd characteristic. We refer to [5, 6]. On the other hand, Ahmadi and Menezes [7] estimate the number of trace-one elements on the trinomial and pentanomial bases for a fast and low-cost implementation of trace computation. They also present a table of irreducible pentanomials whose corresponding polynomial bases have exactly one trace-one element. Each pentanomial of even degree in this table is of the form , where , and exactly one of ,？？and？？ is odd. In this work, we characterize the parity of the number of irreducible factors of this pentanomial. We describe some preliminary results related to Swan-like results in Section 2 and determine the reducibility of the pentanomial mentioned above in Section 3. 2. Preliminaries In this section, we recall Swan’s theorem determining the parity of the number of irreducible factors of a polynomial over and some results about the discriminant and the resultant of polynomials. Let be a field and let , where are the roots of in an extension of . The discriminant of is defined by From the definition, it is clear that has a repeated

Abstract:
The nestling diet of the Fairy Pitta (pitta nympha) was studied by videotaping during breeding period in Jeju Island, 2009. Earthworms of several species were the most common food resources for nestlings, averaging 82% of all items, followed by 4% of Homoptera larvae. The remaining was only rarely recorded, including Lepidopteran larvae and adults, slugs, spiders, beetle adults and larvae (Coleoptera) and grasshoppers. Adults provided the number of preys ranging from 1 to 7 items to chicks per one visit. The average value of prey number per visit was 3.0 (SD = 1.38). The estimated average length of prey was 5.7 cm (SD = 2.85), and 96% of the food items were smaller than 10 cm. The staying time for feeding in an early stage was longer than other stages. Provision rate at a forenoon (mean ± SD, 14.7 ± 4.92) and an afternoon time (15.8 ± 5.30) was not significantly higher than that of noon time (11.7 ± 4.49). These results provide basic information for conservation action of international endangered species of this species.

Abstract:
We find a formula for the number of permutation polynomials of degree q-2 over a finite field Fq, which has q elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree q-2 over a finite field Fq, using the permanent of a matrix whose entries are pth roots of unity and using this obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree d less than q-2.

Abstract:
A. Silverberg (IEEE Trans. Inform. Theory 49, 2003) proposed a question on the equivalence of identifiable parent property and traceability property for Reed-Solomon code family. Earlier studies on Silverberg's problem motivate us to think of the stronger version of the question on equivalence of separation and traceability properties. Both, however, still remain open. In this article, we integrate all the previous works on this problem with an algebraic way, and present some new results. It is notable that the concept of subspace subcode of Reed-Solomon code, which was introduced in error-correcting code theory, provides an interesting prospect for our topic.

Abstract:
In this paper we investigate the separation properties and related bounds of some codes. We tried to obtain a new existence result for $(w_1, w_2)$-separating codes and discuss the "optimality" of the upper bounds. Next we tried to study some interesting relationship between separation and existence of non-trivial subspace subcodes for Reed-Solomon codes.

Abstract:
We study the Gauss map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map $G$ satisfying $\square G=AG$ for some $3\times3$ matrix $A$ are the planes, right circular cones, circular cylinders and spheres.

Abstract:
In this paper, We study upper bounds for separating codes. First, some new upper bound for restricted separating codes is proposed. Finally, we illustrate that the Upper Bound Conjecture for separating Reed-Solomon codes inherited from Silverberg’s question holds true for almost all Reed-Solomon codes.

Abstract:
Separating codes have their applications in collusion-secure fingerprinting for generic digital data, while they are also related to the other structures including hash family, intersection code and group testing. In this paper we study upper bounds for separating codes. First, some new upper bound for restricted separating codes is proposed. Then we illustrate that the Upper Bound Conjecture for separating Reed-Solomon codes inherited from Silverberg's question holds true for almost all Reed-Solomon codes.

Abstract:
Divisibility of trinomials by given polynomials over finite fields has been studied and used to construct orthogonal arrays in recent literature. Dewar et al.\ (Des.\ Codes Cryptogr.\ 45:1-17, 2007) studied the division of trinomials by a given pentanomial over $\F_2$ to obtain the orthogonal arrays of strength at least 3, and finalized their paper with some open questions. One of these questions is concerned with generalizations to the polynomials with more than five terms. In this paper, we consider the divisibility of trinomials by a given maximum weight polynomial over $\F_2$ and apply the result to the construction of the orthogonal arrays of strength at least 3.