ABSTRACT
A method for constructing rotated -lattices, with n a power of , based on totally real subfields of the cyclotomic field , where is an integer, is presented. Lattices exhibiting full diversity in some dimensions not previously addressed are obtained.
Key words
lattices; cyclotomic fields; modulation design; fading channels; minimum product distance
RESUMO
Um método para construir -reticulados rotacionados, com uma potência de , via subcorpos totalmente reais do corpo ciclotômico , onde é um inteiro, é apresentado. Reticulados que exibem diversidade completa em algumas dimensões não abordadas anteriormente são obtidos.
Palavras-chave
reticulados; corpos cyclotômicos; modulação; canais de desvanecimento; distância produto mínima
1 INTRODUCTION
Ring theory and algebric number theory have long shown to be useful tools in the theory of information and coding [8][2] A. Ansari, T. Shah, Z.u. Rahman & A.A. Andrade. Sequences of Primitive and Non-primitive BCH Codes. TEMA (S˜ao Carlos), 19(2) (2018), 369–389. and [2][3] V. Bautista-Ancona, J. Uc-Kuk et al. The discriminant of abelian number fields. Rocky Mountain Journal of Mathematics, 47(1) (2017), 39–52.. In particular, lattices (discrete subgroups of the Euclidean -space ) have played a relevant role in code design for different types of channels, see for example [12][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714., [5][5] E. Bayer-Fluckiger, F. Oggier & E. Viterbo. Algebraic lattice constellations: Bounds on performance. IEEE Transactions on Information Theory, 52(1) (2005), 319–327., [6][6] J. Boutros, E. Viterbo, C. Rastello & J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Transactions on Information Theory, 42(2) (1996), 502–518., and [9][9] P. Elia, B.A. Sethuraman & P.V. Kumar. Perfect Space-Time Codes for Any Number of Antennas. IEEE Trans. Information Theory, 53(11) (2007), 3853–3868.. One central problem in the design of signal constellations for fading channels is to construct lattices from totally real number fields with maximal minimum product distance. Using number-theoretic methods, Andrade et al. [1][1] A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331. and Bayer-Fluckiger et al. [12][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714. presented constructions of algebraic lattices with full diversity and gave closed-form expressions for their minimum product distance using the corresponding algebraic properties.
The -dimensional integer lattice, denoted by , consists of the set of points in whose coordinates are all integers. In [1][1] A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331., for any integer , rotated -lattices, , were constructed from , the maximal real subfield of , where is a primitive -th root of unity. In this work, we extend the method in [1][1] A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331. by considering a particular subfield of , namely, , to construct lattices in dimensions that are powers of .
This paper is organized as follows: In Section 2, notions and results from algebraic number theory that are used in the work are reviewed. In Section 3, rotated -lattices constructed from the totally real fields , with , are presented and their minimum product distances are computed. In Section 4, the concluding remarks are drawn.
2 NUMBER FIELDS BACKGROUND
If be a number field of degree (notation: ), then , for some , which is a root of a monic irreducible polynomial . The distinct roots of , namely, , are the conjugates of . The embeddings of in are the field homomorphisms given by and for all , for . The latter set of embeddings is denoted by . If , for , we say that is totally real. The set is a group under composition, called the Galois group of over and denoted by . The norm and the trace of an element are defined, respectively, as the rational numbers
The set is a ring, called the ring of integers of and denoted by ; moreover, the latter is a -module and it has a basis over , called an integral basis for . The discriminant of , denoted by , is the rational integer given by .Theorem 1. [14, Ch. 2][14] L. Washington. “Introduction to Cyclotomic Fields”. Springer-Verlag, New York, 2 ed. (1997). If , with , then
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, where denotes Euler’s totient function;
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and is an integral basis for ;
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and , where is the maximal real subfield of ;
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and is an integral basis for .
Corollary 2.The degree of over equals , with .Proof. Observe that and use part 3 of Theorem 1.
Corollary 3. If , with , then
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the ring of integers of , namely, , is given by ;
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is an integral basis for , with .
Proof. Observe that and use part 4 of Theorem 1.
Proposition 4. [[13, Theorem 9.12, p. 364][13] K.H. Rosen. “Elementary Number Theory and its Applications”. Addison-Wesley, Reading, MA, 6 ed. (2011). The order of modulo is .
is isomorphic to and is of order [14, Theorem 2.5, p. 11].[14] L. Washington. “Introduction to Cyclotomic Fields”. Springer-Verlag, New York, 2 ed. (1997).. For any odd integer , let be the automorphism of given by . Furthermore, let denote the cyclic subgroup of generated by . The fixed field of is . Observe that is the imaginary unit and so the fixed field of is . With the latter two observations in mind, refer to Figure 1, where the group indicated along each line represents the Galois group of the respective field extension. By Galois theory and more specifically, by [10, Theorem 1.1, Ch. VI][10] S. Lang. “Algebra. Revised third edition, Corrected forth printing”. Graduate Texts in Mathematics, 3 ed. (2003)., and ; furthermore, by [10, Theorem 1.12, Ch. VI][10] S. Lang. “Algebra. Revised third edition, Corrected forth printing”. Graduate Texts in Mathematics, 3 ed. (2003)., it follows that . Lastly, by [10, Theorem 1.14, Ch. VI][10] S. Lang. “Algebra. Revised third edition, Corrected forth printing”. Graduate Texts in Mathematics, 3 ed. (2003)., it follows that .
3 CONSTRUCTION OF IDEAL LATTICES
Let be a totally real number field of degree . An ideal lattice is a lattice , where is an ideal,
and is totally positive, that is, , for all . If is a -basis for , then the generator matrix of is given by The Gram matrix of is given by , where denotes the transpose of .Let be an element of . The product distance of from the origin is defined as
and the minimum product distance of is defined as When is a principal ideal of , the minimum product distance of is given by where , see [12, Theorem 1][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714..Let and be two ideals in . Lattices and are said to be isomorphic (notation: ) [4][4] E. Bayer-Fluckiger. Lattices and number fields. volume 241. Am. Math. Soc. (1999). if there exists such that and , for all .
3.1 Construction from the subfield
Let be the cyclotomic field and , where and where . Throughout this section, let . From [3, Theorem 2.2][3] V. Bautista-Ancona, J. Uc-Kuk et al. The discriminant of abelian number fields. Rocky Mountain Journal of Mathematics, 47(1) (2017), 39–52., one has . The lattices in this section will be built from the ring of integers of , whose an integral basis is given by .
Let be an ideal lattice and a positive integer. Since the Gram matrix of is , a necessary but not sufficient condition for to be isomorphic to , a scaled version of , is that , see [4[4] E. Bayer-Fluckiger. Lattices and number fields. volume 241. Am. Math. Soc. (1999).,12][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714.. Thus, the first step when verifying whether is to find such that is a perfect th power. Since
one has . Using the transitivity of the norm, it follows that Thus is an element of of norm .Proposition 1. [1][1] A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331. If , then
Proposition 2. [1][1] A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331. Let , and , for .
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If , then
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If , then
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If , and , then
Corollary 3. If , then the matrix of in the basis is given by
Proof. It follows directly by Proposition 2.
Matrix of Corollary 3 is the Gram matrix of a rotated -lattice relative to the basis , where , for , and is the canonical basis of Thus , for , is an isomorphism of the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by , for . Hence, it follows the following result.
Proposition 4. Notation as above, if is a -basis for , where , for , then
i.e., the lattice is isomorphic to .Let be the Galois group of over . Thus, the generator matrix of the lattice associated to the ring of integers of is given by
Let The generator matrix of the rotated -lattice is given by see [12, p. 705][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714..Example 3.1. Let be the cyclotomic field and its maximal real subfield . In this case, , and . Considering the -basis for and , it follows that the matrix of is given by
Matrix is the Gram matrix of the rotated -lattice relative to the basis with and , where is the canonical basis of . This implies that , for , is an isomorphism of the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by , for , i.e., and . Therefore, , i.e., the lattice is isomorphic to .Example 3.2. Let be the cyclotomic field and its maximal real subfield . In this case, , and . Considering the -basis for and , it follows that the matrix of is given by
The matrix is the Gram matrix of the rotated -lattice relative to the basis , with and , where is the canonical basis of . This implies that , for , is an isomorphism on the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by , for , i.e., and . Therefore, , i.e., the lattice is isomorphic to .3.2 Construction from the subfield
Let , , , where , and . Throughout this section, let . Thus , , and , where , is a cyclic group isomorphic to . Let and . From Proposition 4, it follows that , i.e., , where is an odd positive integer. Thus, , and therefore, . So,
The lattices are built via the ring of integers of , a real subfield of , whose an integral basis is given by . Since
it follows that . Using the transitivity of the norm, it follows that Thus, is an element of whose norm is equal to .Proposition 5. Let and , for .
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If , then
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If , then
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If , with , then
Proof. By the transitivity of the trace, it follows that . Since and , it follows that
If , for , then Since , it follows that . Thus, Now, for , it follows that . Thus, Since and , it follows that Finally, let , with . Since , with , it follows that Since and , it follows that which proves the proposition.Corollary 6. If , then the matrix of in the basis is given by
Proof. It follows directly from Proposition 5.
Matrix of Corollary 6 is the Gram matrix of a rotated -lattice related to the basis , where , for , and is the canonical basis of . Thus , for , is an isomorphism on the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by for . Hence, one has the following result.
Proposition 7. If , where , for , is a basis of , then
i.e., the lattice is isomorphic to .Let be the Galois group of over . Thus, the generator matrix of the lattice associated to the ring of integers of is given by
Let As before, the generator matrix of the rotated -lattice is then given byExample 3.3. Let be the cyclotomic field and its real subfield given by . In this case, ,
and . Considering the -basis for , namely, , where and , and , it follows that the matrix of is given by Matrix is the Gram matrix of the rotated -lattice relative to the basis with and , where is the canonical basis of . This implies that , for , is an isomorphism of the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by , for , i.e., and . Therefore, , i.e., the lattice is isomorphic to .Example 3.4. Let be the cyclotomic field and its real subfield given by . In this case, ,
and . Considering the basis , where , , and , of and , it follows that the matrix of is given by Matrix is the Gram matrix of the rotated -lattice relative to the basis , with , , and , where is the canonical basis of . This implies that , for , is an isomorphism of the -lattice. The basis which corresponds to the canonical basis of through this isomorphism is then given by , for , i.e., , , and . Therefore, , i.e., the lattice is isomorphic to .From [12, Theorem 1][12] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714., it follows that the minimum product distance of is given by
To compare lattices in different dimensions, we use the parameter . In the next table, we list the minimum product distance of for several dimensions. The entries in the column labeled “bound” were calculated from the minimal discriminant of Abelian and totally real number fields of degree , [11][11] A.M. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Journal de théorie des nombres de Bordeaux, 2(1) (1990), 119–141.. For , the minima appear to be currently unknown, which explains the missing entries.4 CONCLUSION
A method for constructing rotated -lattices via the ring of integers of the subfield of the cyclotomic field with has been presented. The dimensions and were addressed in [7][7] M.O. Damen, K. Abed-Meraim & J.C. Belfiore. Diagonal algebraic space-time block codes. IEEE Transactions on Information Theory, 48(3) (2002), 628–636. using the field , and they have the same as our cyclotomic construction. The lattices presented in this work are all ideal lattices, which allowed us to easily evaluate their minimum product distances from field discriminants, just as in [5][5] E. Bayer-Fluckiger, F. Oggier & E. Viterbo. Algebraic lattice constellations: Bounds on performance. IEEE Transactions on Information Theory, 52(1) (2005), 319–327. and [9][9] P. Elia, B.A. Sethuraman & P.V. Kumar. Perfect Space-Time Codes for Any Number of Antennas. IEEE Trans. Information Theory, 53(11) (2007), 3853–3868..
Acknowledgment
The authors thank the reviewer for carefully reading the manuscript and for all the suggestions that improved the presentation of the work. They also thank FAPESP for its financial support 2013/25977-7.
REFERENCES
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[1]A.A. Andrade, C. Alves & T.B. Carlos. Rotated lattices via the cyclotomic field Q(z2r ). Internat. J. Appl. Math., 19 (2006), 321–331.
-
[2]A. Ansari, T. Shah, Z.u. Rahman & A.A. Andrade. Sequences of Primitive and Non-primitive BCH Codes. TEMA (S˜ao Carlos), 19(2) (2018), 369–389.
-
[3]V. Bautista-Ancona, J. Uc-Kuk et al. The discriminant of abelian number fields. Rocky Mountain Journal of Mathematics, 47(1) (2017), 39–52.
-
[4]E. Bayer-Fluckiger. Lattices and number fields. volume 241. Am. Math. Soc. (1999).
-
[5]E. Bayer-Fluckiger, F. Oggier & E. Viterbo. Algebraic lattice constellations: Bounds on performance. IEEE Transactions on Information Theory, 52(1) (2005), 319–327.
-
[6]J. Boutros, E. Viterbo, C. Rastello & J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Transactions on Information Theory, 42(2) (1996), 502–518.
-
[7]M.O. Damen, K. Abed-Meraim & J.C. Belfiore. Diagonal algebraic space-time block codes. IEEE Transactions on Information Theory, 48(3) (2002), 628–636.
-
[8]A.A. de Andrade, T. Shah & A. Khan. A note on linear codes over semigroup rings. Trends in Applied and Computational Mathematics, 12(2) (2011), 79–89.
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[9]P. Elia, B.A. Sethuraman & P.V. Kumar. Perfect Space-Time Codes for Any Number of Antennas. IEEE Trans. Information Theory, 53(11) (2007), 3853–3868.
-
[10]S. Lang. “Algebra. Revised third edition, Corrected forth printing”. Graduate Texts in Mathematics, 3 ed. (2003).
-
[11]A.M. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Journal de théorie des nombres de Bordeaux, 2(1) (1990), 119–141.
-
[12]F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic Zn-lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information Theory Workshop (Cat. No. 03EX674)”. IEEE (2004), pp. 702–714.
-
[13]K.H. Rosen. “Elementary Number Theory and its Applications”. Addison-Wesley, Reading, MA, 6 ed. (2011).
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[14]L. Washington. “Introduction to Cyclotomic Fields”. Springer-Verlag, New York, 2 ed. (1997).
Publication Dates
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Publication in this collection
13 Dec 2019 -
Date of issue
Sep-Dec 2019
History
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Received
20 Sept 2018 -
Accepted
14 Mar 2019