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Register a new userFive Constructions of Permutation Polynomials over $gf(q^2)$
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Added by
Cunsheng Ding
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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Four recursive constructions of permutation polynomials over $gf(q^2)$ with those over $gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over $gf(q^{2^ell})$ for any positive integer $ell$ with any given permutation polynomial over $gf(q)$. A generic construction of permutation polynomials over $gf(2^{2m})$ with o-polynomials over $gf(2^m)$ is also presented, and a number of new classes of permutation polynomials over $gf(2^{2m})$ are obtained.
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Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, etc. In this paper, we construct several new classes of permutation polynomials, and the necessities of some permutation polynomials are studied.
BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.
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The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length $2^{m}$ by Li textsl{et al.} (the generalized cases I-III for $qge 2$) in 2010 and Liu textsl{et al.} (the generalized cases IV-V for $qge 3$) in 2013 respectively. Those sequences are presented as the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$, we proposed two new constructions of $4^q $-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs (including the generalized cases IV-V) are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. For $q=q_{1}times q_{2}times dotstimes q_{t}$ ($q_k>1$), the number of new offsets in our first construction is lower bounded by a polynomial of $m$ with degree $t$, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial of $m$ and a quadratic polynomial of $m$, respectively. In particular, the numbers of new offsets in our first construction is seven times more than that in the generalized cases IV-V for $q=4$. We also show that the numbers of new offsets in our two constructions is lower bounded by a cubic polynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentioned GCSs over QAM in this paper can be regarded as projections of Golay complementary arrays of size $2times2timescdotstimes2$.
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