DOCSLIB.ORG

Further Results on Some Classes of Permutation Polynomials Over Finite

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Further Results on Some Classes of Permutation Polynomials Over Finite 1 Further results on some classes of permutation polynomials over finite fields Xiaogang Liu Abstract Let Fq denote the finite field with q elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with their studies, and get some further results about the permutation properties of the permutation polynomials. Also, some new classes of permutation polynomials are constructed. For these, we alter the coefficients, exponents or the underlying fields, etc. Index Terms Finite field; Permutation polynomial; AGW criterion; Trace function I. INTRODUCTION F F∗ For a prime power q, let q denote the finite field of order q, and q the multiplicative group. A polynomial f(x) ∈ Fq is called a permutation polynomial (PP) over Fq, if the associated polynomial mapping f : c → f(c) is a permutation of Fq. In recent years, permutation polynomials have become an interesting area of research. They have applications in coding theroy, cryptography and combinatorial design theory [6, 19]. Permutation polynomials with few terms attract many authors’ attention for their simple algebraic structures. In particular, there are many results about permutation binomials and trinomials [2, 5, 13, 18]. For recent achievements on the study of permutation polynomials, the reader can consult [5, 8, 13, 15–17, 21], and the references therein.. Fqn k F Permutation polynomials of the form x+ q (x ) have been studied for special γ ∈ qn , with even characteristic arXiv:1907.03386v1 [cs.IT] 8 Jul 2019 [3, 4], and for the case n =2, i.e., trinomial permutations [9, 13]. Later, Zheng, Yuan and Yu investigated permutation polynomials with the form cx − xs + xqs over Fqn , where s is a positive integer, and c ∈ Fq [22]. For this, they used the well-known result (Lemma 3) to prove three classes of permutation trinomials. For the fourth class of permutation trinomials, symbolic computation method related with resultants was used. And, they found a new relation with a class of PPs of the form k (xq − x + δ)s + cx This work is supported in part by ”Funding for scientific research start-up” of Nanjing Tech University. X. Liu is with College of Computer Science and Technology, Nanjing Tech University, Nanjing City, Jiangsu Province, PR China 211800 e-mail:[email protected]. 2 over Fqn , where δ is arbitrary and c ∈ Fq. This class of permutation polynomials are related to δ. Based on their relationship, the aforementioned class of pemutation trinomials without restriction on δ are derived. Complete permutation polynomial (CPP) is a permutation polynomial such that f(x)+ x is also a permutation polynomial. CPPs were introduced with the construction of Latin squares [10]. In the study of CPPs, Li etc., k found that certain polynomials over F2n can have the same permutation properties as ax + bx over F2m for n positive divisor m of n such that m is odd [11]. They constructed some permutation binomials, and thus more n k n F∗ permutation polynomials of the form a[Trm(x)] +u(c+x)(Trm(x)+x)+bx can be obtained, here a,b,c,u ∈ 2m F pm pm are constants. They also studied permutation polynomials over p2m of the form ax + bx + h(x ± x), and F Niho-type permutation trinomials over p2m were constructed. In this paper, we revisit the permutation polynomials proposed in [11, 22]. Based on their results we make some modifications on the condtions, and get some further results which are generalizations. Also, we construct some new classes of permutation polynomials in Sections II and III. Before going on, let us present the following lemmas, which might be useful for our study. ∗ F F m Lemma 1: [22] Let m, k be integers with 0 <k<m and l = gcd(k,m). Let c ∈ ql and g(x) ∈ q [x]. Then qk qk g(x − x + δ)+ cx permutes Fqm for each δ ∈ Fqm if and only if h(x)= g(x) − g(x)+ cx permutes Fqm . Using the same idea as in [22, Proposition 3], we can get the following lemma. ∗ F F m Lemma 2: Let m, k be integers with 0 <k<m and l = gcd(k,m). Let c ∈ ql and g(x) ∈ q [x]. Then qk qk g(x + x + δ)+ cx permutes Fqm for each δ ∈ Fqm if and only if h(x)= g(x) + g(x)+ cx permutes Fqm . r (q−1)/d Lemma 3: [23] Let d,r > 0 with d | q − 1, and let h(x) ∈ Fq[x]. Then f(x) = x h(x ) permutes Fq if and only if the following two conditions hold: (i) gcd(r, (q − 1)/d) = 1; r (q−1)/d (ii) x h(x) permutes µd, where µd denotes the d-th root of unity in Fq. F∗ n b Lemma 4: [20] For a positive integer m, and a,b ∈ 22m satisfying Tr1 ( a2 )=0. Then the quadratic equation x2 + ax + b =0 has (i) both solutions in the unit circle if and only if a m b m 1 b = a2m and Tr1 ( a2 )= Tr1 ( a2m+1 )=1; (ii) exactly on solution in the unit circle if and only if a 2m+1 2m+1 2m+1 2 2m 2m+1 b 6= a2m and (1 + b )(1 + a + b )+ a b + a b =0. II. PERMUTATION POLYNOMIALS WITH TWO OR THREE TERMS In this section, we study the permutation behavior of four kinds of polynomials. Two kinds of them have three terms, or to some extent are related with a three term polynomial, see Corollary 1 and Proposition 2. Two kinds of them have two terms, see Proposition 3 and Proposition 4. They come from transformations from permutation polynomials proposed in [11, 22]. F F∗ Proposition 1: For a positive integer m, a fixed δ ∈ 23m and c ∈ 2m , the polynomial m 2m f(x) = (x2 + x + δ)2 +1 + cx 3 F is a permutation of q3m . Proof: [12, Proposition 4] says that m 2m g(x) = (x2 + x + δ)2 +1 + x F F∗ is a permutation of q3m . Set x = c0y, with c0 ∈ 2m . Then we have 2m 22m+1 g(x)= g(c0y) = ((c0y) + c0y + δ) + c0y m 2m = c2(y2 + y + 1 δ)2 +1 + c y 0 c0 0 m 2m = c2((y2 + y + 1 δ)2 +1 + 1 y). 0 c0 c0 1 Since g(x) is a permutation for every δ ∈ F 3m , setting δ0 = δ, q c0 2m 22m+1 1 f0(y) = (y + y + δ0) + y c0 1 F m F m is a permutation of q3 for every δ0 ∈ q3 . Let c0 = c , 2m 22m+1 f(y) = (y + y + δ0) + cy F F is a permutation of q3m for every δ0 ∈ q3m . Note that in Proposition 1, c is unconnected with δ. Combing with Lemma 1, the following result follows. F∗ Corollary 1: For a positive integer m and c ∈ 2m , the polynomial m 2m 2m g(x)= x2 (2 +1) + x2 +1 + cx F is a permutation of 23m . Example 1: Set m =3. Using Magma, it can be verified that the following polynomial g(x)= x520 + x65 + cx F F∗ is a permutation over 512 for c ∈ 8. In Theorem 10 [22], permutation polynomials of the kind x + xs + xqs are proposed, but with all coefficients 1, q+1 here we add a variable to the coefficient to investigate more general situations. Denote µq+1 = {x|x =1}. m q2+q+1 Proposition 2: Let q =2 be even with q ≡ 1 (mod 3), and s = 3 . Then f(x)= cx + xs + cqxqs F F∗ m q+1 is a permutation polynomial over q2 for c ∈ q2 satisfying Tr1 (c )=0. Proof: It can be found that 3|(q + 2) from the assumption q ≡ 1 (mod 3). Let us rewrite the polynomial f(x) in the following q+2 (q−1)+1 q ( q(q+2) +1)(q−1)+1 f(x)= cx + x 3 + c x 3 . (1) q+2 Set u = 3 , equation (1) becomes − − f(x)= x(c + x(q 1)u + cqx(qu+1)(q 1)). By Lemma 3, f(x) is a permutation polynomial if and only if − g(x)= x(c + xu + cqxqu+1)q 1 4 3u q+2 permutes µq+1. On the set µq+1, x = x = x and g(x) becomes − − g(x)= x(c + xu + cqx u+1)q 1. It is necessary to show that − c + xu + cqx u+1 =0 has no zeros on µq+1. The above equation can be written as c + xu + cqx2u =0. (2) q+2 u We can find that gcd(u, q +1)= gcd( 3 , q +1)=1. Set z = x , then equation (2) becomes 1 c z2 + z + =0. (3) cq cq b q+1 F For Lemma 4, we have a2 = c which lies in q. And equation (3) has no solutions in the unit circle by our assumption. Let us make the following transformations for g(x) on the unit circle µq+1, − − cq +x u+cx 1+u g(x) = x c+xu+cq x−u+1 − cq x+x1 u+cxu = c+xu+cq x−u+1 cq x3u+x2u+cxu = c+xu+cqx2u u cq x2u+xu+c = x c+xu+cqx2u = xu. Thus, g(x) is a permutation of µq+1. Example 2: Set m =4. Using Magma, it can be verified that f(x)= cx + x91 + c16x1456 F 4 17 is a permutation polynomial over 256 for Tr1(c )=0.
Load more

Recommended publications
  • ON SOME PERMUTATION POLYNOMIALS OVER Fq of THE
    ON SOME PERMUTATION POLYNOMIALS OVER Fq r (q 1)=d OF THE FORM x h(x − ) MICHAEL E. ZIEVE Abstract. Several recent papers have given criteria for certain polynomials to permute Fq, in terms of the periods of certain gen- eralized Lucas sequences. We show that these results follow from a more general criterion which does not involve such sequences. 1. Introduction A polynomial over a finite field is called a permutation polynomial if it permutes the elements of the field. These polynomials first arose in work of Betti [5], Mathieu [25] and Hermite [20] as a way to represent permutations. A general theory was developed by Hermite [20] and Dickson [13], with many subsequent developments by Carlitz and oth- ers. The study of permutation polynomials has intensified in the past few decades, due both to various applications (e.g., [8, 11, 14, 30]) and to an increasing appreciation of the depth of the subtleties inherent to permutation polynomials themselves (for instance, work on permuta- tion polynomials led to a bound on the automorphism group of a curve with ordinary Jacobian [19]). The interesting aspect of permutation polynomials is the interplay between two different ways of representing an object: combinatorially, as a mapping permuting a set, and algebraically, as a polynomial. This is exemplified by one of the first results in the subject, namely that there is no permutation polynomial over Fq of degree q 1 if q > 2 [20]. Much recent work has focused on low-degree permutation− polynomials, as these have quite remarkable properties: for instance, a polynomial of 1=4 degree at most q which permutes Fq will automatically permute Fqn for infinitely many n.
    [Show full text]
  • Constructing Permutation Arrays from Semilinear Groups
    Constructing Permutation Arrays from Semilinear Groups Avi Levy 2/17/2013 Contents 1 Introduction 2 2 Overview and Notation 2 2.1 Finite Fields . 2 2.2 Permutation Polynomials . 2 2.3 Classical Groups . 2 2.4 Hamming distance and Permutation Arrays . 3 3 Constructions 3 3.1 AGL(1; Fq) ............................ 3 3.2 AΓL(1; Fq)............................. 3 3.3 P GL(2; Fq) ............................ 4 3.4 P ΓL(1; Fq)............................. 4 4 Hamming Distance Computations 4 4.1 Preliminaries . 4 4.2 AΓL ................................ 5 4.3 P ΓL ................................ 6 5 Root-counting Results 7 6 Conclusions 10 7 Bibliography 10 1 1 Introduction This is a preliminary writeup of my work with the semilinear groups. These groups yield new permutation arrays with large pairwise Hamming distances. Their properties are well-understood, which allows the minimal pairwise Hamming distance to be obtained without resorting to direct computation. 2 Overview and Notation 2.1 Finite Fields n Given q = p , a power of a prime, call the unique field with q elements Fq. 2 q−2 Fq = f0; 1; g; g ; ··· ; g g where g is any generator of Fq. 2.2 Permutation Polynomials Let f : Fq ! Fq be a polynomial function with coefficients in Fq. If f is one-to-one, then f permutes the elements of Fq. In this case, f is called a permutation polynomial, and the permutation corresponding to f is: σf : x ! f(x) 2.3 Classical Groups There are four families of groups that we are concerned with in this paper. • AGL(n; F): affine general linear group of dimension n over the field F.
    [Show full text]
  • New Permutation Trinomials from Niho Exponents Over Finite Fields
    New Permutation Trinomials From Niho Exponents over Finite Fields with Even Characteristic Nian Li and Tor Helleseth ∗ Abstract In this paper, a class of permutation trinomials of Niho type over finite fields with even char- acteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works. 1 Introduction m Let p be a prime and Fpm denote the finite field with p elements. A polynomial f(x) ∈ Fpm [x] is called a permutation polynomial of Fpm if the associated polynomial function f : c 7→ f(c) from Fpm to Fpm is a permutation of Fpm [15]. Permutation polynomials over finite fields, which were first studied by Hermite [6] and Dickson [2], have wide applications in the areas of mathematics and engineering such as coding theory, cryptography and combinatorial designs. The construction of permutation polynomials with either a simple form or certain desired property is an interesting research problem. For some recent constructions of permutation polynomials with a simple form, the reader is referred to [3, 8, 10, 13, 18, 19, 23] and for some constructions of permutation polynomials with certain desired property, the reader is referred to [1, 4, 5, 12, 20, 21, 22] and the references therein for example. Permutation binomials and trinomials attract researchers’ attention in arXiv:1606.03768v1 [cs.IT] 12 Jun 2016 recent years due to their simple algebraic form, and some nice works on them had been achieved in [3, 10, 18, 23].
    [Show full text]
  • New Permutation Trinomials Constructed from Fractional
    1 New Permutation Trinomials Constructed from Fractional Polynomials Kangquan Li, Longjiang Qu, Chao Li and Shaojing Fu** Abstract Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures. Index Terms Finite Field, Permutation Trinomial, Fractional Polynomial. 1. INTRODUCTION Let Fq be the finite field with q elements. A polynomial f ∈ Fq[x] is called a permutation polynomial (PP) if the induced mapping x → f(x) is a permutation of Fq. The study about permutation polynomials over finite fields attracts people’s interest for many years due to their wide applications in coding theory [15, 24], cryptography [19, 22, 23] and combinatorial designs [5]. For example, Ding et al.[5] constructed a family of skew Hadamard difference sets via the Dickson permutation polynomial of order five, which disproved the longstanding conjecture on skew Hadamard difference sets. More recent progress on permutation polynomials can be found in [2, 4, 9, 10, 12–14, 21, 25, 27]. Permutation trinomials over finite fields are in particular interesting for their simple algebraic form and arXiv:1605.06216v1 [cs.IT] 20 May 2016 additional extraordinary properties.
    [Show full text]
  • Polynomial Functions and Permutation Polynomials Over Some Finite Commutative Rings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Number Theory 105 (2004) 192–202 http://www.elsevier.com/locate/jnt Polynomial functions and permutation polynomials over some finite commutative rings$ Qifan Zhang College of Mathematics, Sichuan University, Chengdu 610064, China Received 18 July 2003 Communicated by S.-W. Zhang Abstract We extend some classical results on polynomial functions mod pl: We prove all results in algebraic methods avoiding any combinatorial calculation. As applications of our methods, we obtain some interesting new results on permutation polynomials in several variables over some finite commutative rings. r 2003 Elsevier Inc. All rights reserved. Keywords: Polynomial function; Permutation polynomial; Witt polynomial; Teichmu¨ ller element 1. Introduction We are interested in polynomial functions and permutation polynomials over finite commutative rings. Without loss of generality, we need only to consider local rings, for example, the ring Z=plZ: The polynomial functions over Z=plZ are widely used in the literature on finite combinatorics (see e.g. [1,2,11]). It is well-known that each function over a finite field is a polynomial function. One should ask when a function f over Z=plZ is a polynomial function. A clear necessary condition, easily l derived from Taylor’s formula, is that there exist functions fi over Z=p Z; i ¼ 0; 1; y; l À 1 such that for any x; sAZ=plZ; the following equality holds: lÀ1 f ðx þ psÞ¼f0ðxÞþf1ðxÞps þ ? þ flÀ1ðxÞðpsÞ : $This work is supported by NSFC (10128103,19901023).
    [Show full text]
  • Permutation Polynomial Interleavers
    1 Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective Oscar Y. Takeshita Dept. of Electrical and Computer Engineering 2015 Neil Avenue The Ohio State University Columbus, OH 43210 [email protected] Submitted to the IEEE Transactions on Information Theory January 12, 2006 Abstract An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic con- structions are important because they admit analytical designs and simple, practical hardware implementation. The spread factor of an interleaver is a common measure for turbo coding applications. Maximum-spread interleavers are interleavers whose spread factors achieve the upper bound. An infinite sequence of quadratic permutation poly- nomials over integer rings that generate maximum-spread interleavers is presented. New properties of permutation polynomial interleavers are investigated from an algebraic-geometric perspective resulting in a new non-linearity metric for interleavers. A new interleaver metric that is a function of both the non-linearity metric and the spread factor is proposed. It is numerically demonstrated that the spread factor has a diminishing importance with the block length. A table of good interleavers for a variety of interleaver lengths according to the new metric is listed. Extensive computer simulation results with impressive frame error rates confirm the efficacy of the new metric. Further, when tail-biting constituent codes are used, the resulting turbo codes are quasi-cyclic. Index Terms algebraic, geometry, interleaver, permutation polynomial, quadratic, quasi-cyclic, spread, turbo code. arXiv:cs/0601048v1 [cs.IT] 12 Jan 2006 2 I. INTRODUCTION Interleavers for turbo codes [1]–[14] have been extensively investigated. However, the design of inter- leavers for turbo codes is complex enough and we believe there are still several relevant open questions.
    [Show full text]
  • A Classification of Permutation Polynomials Through Some Linear
    A Classification of Permutation Polynomials through some linear maps Megha M. Kolhekar, Harish K. Pillai {meghakolhekar, hp}@ee.iitb.ac.in Department of Electrical Engineering, IIT Bombay India November 12, 2019 Abstract In this paper, we propose linear maps over the space of all polyno- mials f(x) in Fq[x] that map 0 to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with per- mutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the gen- eralized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes of permutation polynomials stud- ied in literature neatly fall into classes defined using these linear maps. We characterize the shapes of permutation polynomials that appear in the various generalized eigenspaces of these linear maps. For the case of Fp, these generalized eigenspaces provide a degree-wise distribution of poly- nomials (and therefore permutation polynomials) over Fp. We show that for Fq, it is sufficient to consider only a few of these linear maps. The intersection of the generalized eigenspaces of these linear maps contain (permutation) polynomials of certain shapes. In this context, we study a F class of permutation polynomials over p2 . We show that the permutation polynomials in this class are closed under compositional inverses. We also do some enumeration of permutation polynomials of certain shapes. arXiv:1911.03689v1 [math.NT] 9 Nov 2019 1 Introduction and Related Work Permutation polynomials (PPs) are important in cryptography, combina- torics and coding theory, to name a few areas.
    [Show full text]
  • Constructions of Rank Modulation Codes That Are Useful for Error Correction in Powerline Communication
    1 Constructions of Rank Modulation Codes Arya Mazumdar∗, Alexander Barg§ and Gilles Z´emora Abstract—Rank modulation is a way of encoding information the drift in different cells may occur at different speed, errors to correct errors in flash memory devices as well as impulse introduced in the data are adequately accounted for by tracking noise in transmission lines. Modeling rank modulation involves the relative value of adjacent cells rather than the absolute construction of packings of the space of permutations equipped with the Kendall tau distance. values of cell charges. Storing information in relative values We present several general constructions of codes in permuta- of the charges also simplifies the rewriting of the data because tions that cover a broad range of code parameters. In particular, we do not need to reach any particular value of the charge as we show a number of ways in which conventional error-correcting long as we have the desired ranking, thereby reducing the risk codes can be modified to correct errors in the Kendall space. of overprogramming. Based on these ideas, Jiang et al. [14], Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to [15] suggested to use the rank modulation scheme for error- correct errors in the Hamming metric. For instance, from binary correcting coding of data in flash memories. A similar noise BCH codes we obtain codes correcting t Kendall errors in n model arises in transmission over channels subject to impulse t memory cells that support the order of n!/(log2 n!) messages, noise that changes the value of the signal substantially but for any constant t = 1, 2,...
    [Show full text]
  • PERMUTATION BINOMIALS OVER FINITE FIELDS 1. Introduction A
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 8, August 2009, Pages 4169–4180 S 0002-9947(09)04578-4 Article electronically published on March 17, 2009 PERMUTATION BINOMIALS OVER FINITE FIELDS ARIANE M. MASUDA AND MICHAEL E. ZIEVE Abstract. m n F We prove that if x + ax permutes the√ prime field p,where ∈ F∗ − − − m>n>0anda p,thengcd(m n, p 1) > p 1. Conversely, we prove that if q ≥ 4andm>n>0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over Fq of the form xm + axn if and only if gcd(m, n, q − 1) = 1. 1. Introduction A polynomial over a finite field is called a permutation polynomial if it permutes the elements of the field. These polynomials first arose in work of Betti [1] and Hermite [10] as a way to represent permutations. A general theory was developed by Hermite [10] and Dickson [6], with many subsequent developments by Carlitz and others. The simplest class of nonconstant polynomials are the monomials m m x with m>0, and one easily checks that x permutes Fq if and only if m is coprime to q − 1. However, already for binomials the situation becomes much more mysterious. Some examples occurred in Hermite’s work [10], and Mathieu [17] pi i showed that x − ax permutes Fq whenever a is not a (p − 1)-th power in Fq;here p denotes the characteristic of Fq. A general nonexistence result was proved by Niederreiter and Robinson [20] and improved by Turnwald [28]: m n F ∈ F∗ Theorem 1.1.
    [Show full text]
  • The Structure of a Group of Permutation Polynomials
    J. Austral. Math. Soc. (Series A) 38 (1985), 164-170 THE STRUCTURE OF A GROUP OF PERMUTATION POLYNOMIALS GARY L. MULLEN and HARALD MEDERREITER (Received 24 November 1983; revised 31 January 1984) Communicated by R. Lidl Abstract Let Gq be the group of permutations of the finite field Fq of odd order q that can be represented by iq+v>/2 polynomials of the form ax + bx with a, b e Fq. It is shown that Gq is isomorphic to the regular wreath product of two cyclic groups. The structure of Gq can also be described in terms of cyclic, dicyclic, and dihedral groups. It also turns out that Gq is isomorphic to the symmetry group of a regular complex polygon. 1980 Mathematics subject classification (Amer. Math. Soc): 12 C 05, 20 B 25, 20 E 22, 51 F 15 1. Introduction Let F be the finite field of order q. Then every mapping from Fq into itself can be uniquely represented by a polynomial in Fq[x] of degree less than q, and composition of mappings corresponds to composition of polynomials mod(jc* — x) (see [9, Chapter 7]). In particular, every group of permutations of Fq can be represented by a set of polynomials in Fq[x] of degree less than q that is closed under composition vao&(xq — x). According to a well-known definition (see [8, Chapter 4], [9, Chapter 7]), a polynomial / over Fq for which the corresponding polynomial mapping c e Fq -* f(c) is a permutation is called a permutation polynomial of Fq. Numerous papers have been written on the structure of permutation groups represented by a given group of permutation polynomials of F under composition mod(x'7 — x); see for example Carlitz [1], > 1985 Australian Mathematical Society 0263-6115/85 $A2.00 + 0.00 164 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • 8.3 Value Sets of Polynomials
    Permutation polynomials 225 arbitrary subfields of the finite field. The results developed in this paper are used to con- struct additional complete sets of frequency squares, rectangles and hyperrectangles, and orthogonal arrays. Some of the theorems present a relationship between permutation poly- nomials of a finite field and field permutation functions, an implicit bound on the possible number of functions in an orthogonal field system, and results related to the generation of orthogonal field systems. 8.2.21 Remark For finite rings there are two concepts to distinguish: permutation polynomials (as above) and strong permutation polynomials. The latter are defined via the cardinality of the inverse image. Polynomials are strong permutation polynomials (or strong orthogonal systems) in n variables if they can be completed to an orthogonal system of n polynomials in n variables. 8.2.22 Theorem [1129] If R is a local ring whose maximal ideal has a minimal number m of generators, then for every n > m there exists a permutation polynomial in n variables that is not a strong permutation polynomial. 8.2.23 Corollary [1129] Every permutation polynomial in any number of variables over a local ring R is strong if and only if R is a finite field. 8.2.24 Remark Wei and Zhang showed [2954] that if n m in the setting of Theorem 8.2.22, then every orthogonal system of k polynomials in n variables≤ can be completed to an orthogonal system of n polynomials (and in particular, every permutation polynomial is strong). See Also 9.4 Considers κ-polynomials used for constructions of semifields.
    [Show full text]
  • Quadratic Permutation Polynomial Interleavers for LTE Turbo Coding
    International Journal of Future Computer and Communication, Vol. 2, No. 4, August 2013 Quadratic Permutation Polynomial Interleavers for LTE Turbo Coding Ching-Lung Chi Abstract—Interleaver is an important component of turbo min δ, (2) , code and has a strong impact on its error correcting performance. Quadratic permutation polynomial (QPP) δ, is the Lee metric between points interleaver is a contention-free interleaver which is suitable for , π and p j,πj parallel turbo decoder implementation. To improve the performance of LTE QPP interleaver, largest-spread and , || | | (3) maximum-spread QPP interleavers are analyzed for Long Term Evolution (LTE) turbo codes in this paper. Compared where with LTE QPP interleaver, those algorithms have low computing complexity and better performances from || mod , mod (4) simulation results. The quadratic polynomials which lead to the largest spreading factor for some interleaver lengths are given in Index Terms—Quadratic permutation polynomial (QPP) interleaver, Long Term Evolution (LTE), largest-spread QPP [2]. An algorithm for faster computation of is also interleaver, maximum-spread QPP interleaver. presented. It is based on the representatives of orbits in the representation of interleavercode. Section II gives a brief review of QPP interleaver. Largest-Spread and Maximum-Spread QPP interleavers are I. INTRODUCTION the topics of Section III. Section IV presents the bit error The selection of turbo coding was considered during the rates (BER) resulted from simulations for LTE standard, LS study phase of Long Term Evolution (LTE) standard to and MS QPP inter leavers length 512. Section V concludes meet the stringent requirements. Following deliberations the paper. within the working group, the quadratic permutation polynomials (QPP) were selected as interleavers for turbo codes, emerging as the most promising solutions to the LTE II.
    [Show full text]