aa r X i v : . [ m a t h . M G ] F e b A NOTE ON THE LOW-DIMENSIONAL MINKOWSKI-REDUCTION ´AKOS G.HORV ´ATH
Abstract.
In this research-expository paper we recall the basic results of reduction theory of positivedefinite quadratic forms. Using the result of Ryskov on admissible centerings and the result of Tammelaabout the determination of a Minkowski-reduced form, we prove that the absolute values of coordinatesof a minimum vector in a six-dimensional Minkowski-reduced basis are less or equal to three. To get thislittle sharpening of the result which can be deduced automatically from Tammela’s works we combinesome elementary geometric reasonings with the mentioned theoretical results. Introduction
A lattice is a discrete subgroup of R n . Any lattice L has a lattice basis, i.e., a set of linearly independentvectors such that the lattice is the set of all integer linear combinations of them. All bases have the samenumber of elements, called the dimension of the lattice. A reduced basis is a basis made of reasonablyshort vectors that are almost orthogonal. There exist many different notions of reduction. The mostimportant classic reductions are of Hermite [13], of Minkowski [16], of Korkine-Zolotarev [14] and ofVenkov [28]. For computational point of view well-usable the Lenstra-Lenstra-Lov´asz (LLL) reduction[15] and there are some nice recent variation as the one of Nguyen and Stehl´e in works [18] and [19].Comparison of these methods in theoretical point of view is not so simple as we think for a first glance.The long story that the reductions of Minkowski and of Hermite are agree up to six dimension and givesdistinct results in higher dimension can be found in [21]. We shortly review the history of this result.Ryskov proved in [20] that these are agree for dimensions n ≤ n ≥
11. In a privateletter Barnes wrote to Ryskov that there is a counting error in his paper from which follows that thetwo reductions are distinct in dimensions 9 and 10. Meanwhile P.P. Tammela proved in [24] that thereduction by Minkowski and by Hermite are equivalent in dimension 6. Finally, Ryskov in [21] closedthe problem, he gave an example of dimension seven for a quadratic form which is Minkowski-reducedbut not Hermite-reduced one. More detailed history of the reduction theory see the book [12] and theliterature therein.Very important task in geometric number theory to find a shortest vector in a lattice which we callminimum vector of the lattice. All classical theory of reduction a priori assumes the solvability of thisproblem. The problem of searching of minimum vectors connects reduction theory of positive definitequadratic forms, the theory of admissible centerings of a minimum parallelepiped and the determinationproblem of the Dirichlet-Voronoi cell of a lattice. The geometric structure of minimum vectors of a latticealso important in another theory of mathematics it can be used in the packing and covering problems ofdiscrete geometry, in coding theory, and the theory of root lattices which is used in modern differentialgeometry.In this expository-research paper we concentrate to the coordinates of minimum vectors in a Minkowski-reduced basis. A definition or result can be found in that section where it first appear. In the first sectionwe recall the definition of Minkowski and Hermite reductions and give a 9-dimensional example for aMinkowski-reduced basis which is not Hermite-reduced one. Its advantage is the simple geometric proofon the basis of the natural geometric definitions.(This example can be found in [8] in Hungarian.) In thesecond section we summarize the theory of admissible centerings and that results which needed to theproof of our Theorem 1. It can be found in the last section of this paper.2.
Minkowski-reduced form which is not Hermite-reduced one.
In our paper n -dimensional lattice L [ e , . . . , e n ] is the set of the integer linear combinations of abasis { e , . . . , e n } of the n -dimensional real vector space R n = Lin[ e , . . . , e n ]. A k -element primitive Mathematics Subject Classification.
Key words and phrases.
Dirichlet-Voronoi cell, Hermite-reduction, minimum vector of a lattice, Minkowski-reduction,admissible centering, positive quadratic forms. system { f , . . . , f k } of the lattice contains such lattice vectors for which Lin[ f , . . . , f k ] ∩ L [ e , . . . , e n ] = L [ f , . . . , f k ]. A basis of the lattice is a primitive system of n elements. A basis of the lattice is reduced byHermite (or Hermite-reduced one) if it is a lexicographical minimum of bases arranged by the increasinglengths of their elements. A Minkowski-reduced basis (or basis reduced by Minkowski) can be get on thefollowing algorithm: Denote by a a shortest vector of the lattice, let a be a shortest, from a linearlyindependent vector, which gives with a a primitive system, and similarly for a i i = 3 , . . . , n . Then { a , . . . , a n } is a basis reduced by Minkowski. Clearly, all subsets of a primitive system are also primitiveone; a shortest vector of a lattice is a primitive one; in all n -lattice there exist bases reduced by Minkowskiand also bases reduced by Hermite. If a basis is reduced by Hermite then it is reduced by Minkowski,and a basis which contains only minimum vectors is reduced by Hermite.In every lattice we can use the following algorithm to get a linearly independent vector system of shortvectors: Denote by a a shortest vector of the lattice, let a be a shortest, from a linearly independentvector and similarly for a i i = 3 , . . . , n . The system { a , . . . , a n } is a so-called system of successiveminimum vectors . The lengths of these vectors are the successive minima of the lattice. In general it isnot a basis in the original lattice.To get a basis in a lattice which is reduced by Minkowski but is not reduced by Hermite consider thefollowing construction. Let the lattice L [ e , . . . , e ] of dimension 9 in the twelve-dimensional Euclideanspace R is spanned by the column e i of the matrix:(1) E =
00 0 1 0 0 0 0
00 0 0 1 0 0 0 0 −
12 13 (The coordinates are written with respect to an orthonormal vector system of R .) We prove that thelength of a lattice vector is greater or equal to 1 so the columns of the matrix E (of lengths 1) forms aHermite-reduced basis of L . Thus it is also a Minkowski-reduced basis (and the corresponding quadraticform is a Minkowski-reduced form).In fact, the square of the lengths of the lattice vector v = P i =1 x i e i can be get in two ways. It is thesquared sum of its coordinates and also an integer substitution of the symmetric quadratic form f ( x )generated by the Gram matrix of the basis { e , . . . , e } . Hence we have(2) X i =1 v i = v T v = x T ( E T E ) x = x T Gx =: f ( x ) = X i =1 x i + 12 x x + x x − x x . From this we can see if x = x = 0 then f ( x ) ≥
1. If x = 0 and x = 0 then x + x x + x ≥ x and x because | x x | ≤ max { x , x } . Similarly, if x = 0 and x = 0 then x + x x + x ≥
1. Finally, if x = 0 and x = 0 then we have three cases.Assume first that | x | ≥
3. Then | v | ≥ | v | ≥
1, too.Secondly, if | x | = 2 and x is an odd integer then | v | , | v | ≥ , | v | , | v | , | v | , | v | ≥ and | v | ≥ showing that | v | ≥ + >
1. Similarly, if | x | = 2 and x is even then | v | , | v | , | v | , | v | , | v ≥ and | v | ≥ implying that | v | ≥ | x | = 1 and again either x is odd or x is even. In both cases | v | , | v | , | v | , | v | , | v | ≥ and either | v | , | v | ≥ or | v ≥ showing the statement.Let denote e ⋆ a unit lattice vector defined by the equality e ⋆ := − e − X i =2 e i + 2 e + 3 e = (0 , , , , , , , , , , , T . Note that the vector system { e , . . . , e , e ⋆ } is a primitive system. Really, if a lattice vector v ∈ L [ e , . . . , e ] is in that linear subspace which is spanned by this linearly independent vectors, then itslast coordinate is zero. This gives the equality 3 x = 2 x , hence x is even and x can be divided by 3.Hence the coordinates of x e + x e are integers moreover the first four coordinates are always agree toeach other. Hence v is in the integer linear hull of { e , . . . , e , e ⋆ } as we stated.Since this primitive system contains only minimal vectors, it can be supplemented to a Minkowski-reduced basis with a vector e ⋆ ∈ L [ e , . . . , e ]. Since this will be a basis its coordinates has to satisfy thefollowing congruences v ≡ v ≡ v ≡ v ≡ ±
16 mod (1) v ≡ v ≡
12 mod (1)and v ≡ v ≡ v ≡ v ≡ v ≡ ± . From this we get that | e ⋆ | ≥
436 + 2036 + 1836 = 76 > . Hence this basis is Minkowski-reduced basis but not Hermite-reduced one. (It can be seen easily thatwith the vector e ⋆ := − e + e + e , the above primitive system is a Minkowski-reduced basis.)3. The admissible centering of a lattice
In general, the lattice L [ e , . . . , e n ] of dimension n is a centering of the lattice L ′ [ a , . . . , a n ] of thesame dimension if L ′ [ a , . . . , a n ] ⊂ L [ e , . . . , e n ]. We have a quite simple possibility to define the conceptof admissible centering of L ′ [ a , . . . , a n ] if in it there is a basis which elements have minimal length (brieflywe say that in L ′ [ a , . . . , a n ] there is a minimum basis ). In this case, we say that the centering is admissible if min L ′ [ a , . . . , a n ] = min L [ e , . . . , e n ], where min L [ e , . . . , e n ] means the minimal value of the lengthsof the vectors of L [ e , . . . , e n ]. Note, that in general there is no minimum basis of a lattice, hence theabove strict idea of permissibility cannot be interpretable. On the other hand, in all lattices we canfind system of successive minimum vectors. In this case take n independent lattice vectors { a , . . . , a n } which lengths are the successive minima of the lattice L ′ [ e , . . . , e n ]. Then L ′ [ e , . . . , e n ] is a centeringof L [ a , . . . , a n ]. We say that this centering is admissible if for arbitrary sequence 1 ≤ i < . . . i k ≤ n ofindices in the lattice Lin[ a i , . . . , a i k ] ∩ L ′ [ e , . . . , e n ] the successive minima are | a i | ≤ . . . ≤ | a i k | .If the lattice L [ a , . . . , a n ] with the minimum basis [ a , . . . , a n ] has a admissible centering L ′ [ e , . . . , e n ]then in the half-closed basic parallelepiped Π[ a , . . . , a n ] are finitely many points { s , . . . s j } of L ′ [ e , . . . , e n ]with the following properties: • The vector system { a , . . . , a n , s , . . . s j } generates precisely the lattice L ′ [ e , . . . , e n ]. • min L [ a , . . . , a n ] = min L ′ [ e , . . . , e n ].The modified version of this denotation for admissible centerings of a lattice with basis containingthe successive minimum vectors of L ′ [ e , . . . , e n ] is also used. Ryskov in [23] proved that for everyadmissible centerings of a parallelepiped based on the successive minimum vectors of a lattice there isan affinity which sends to this parallelepiped to a admissible centerings of a parallelepiped based on asystem of minimum vectors of the corresponding lattice. This result says that the admissible centeringsof the lattices having basis from successive minimum vectors combinatorially the same as the admissiblecenterings of those lattices which have minimum basis. A lattice L ⋆ [ e , . . . , e n ] (given by a basis) is alwaysa admissible centerings of a lattice L [ a , . . . , a n ] which is spanned by a system of successive minimumvectors of L ′ [ e , . . . , e n ]. By Ryskov’s result the relative connection between the two lattices always can bedescribed as a admissible centering of a lattice with minimum basis. If a given dimension the admissiblecenterings are classified then we know the possible connections between the given lattice L ′ [ e , . . . , e n ] andits sublattices L [ a , . . . , a n ]. The following table of Ryskov contains the most important geometric data ofthe admissible centerings of a minimum parallelepiped. To review it note that the above points { s , . . . s j } of the parallelepiped Π[ a , . . . , a n ] have rational coordinates with respect to the basis { a , . . . , a n } . Infact, if a basis of L ′ [ e , . . . , e n ] is { f , . . . , f n } , then the matrix A of the linear transformation sending f i to e i has integer elements. The coordinates of the points { s , . . . s j } are also integers with respect to thebasis { f , . . . , f n } . Hence the basis change f i to e i changes the coordinates of { s , . . . , s j } such rationalcoordinates whose denominators divide the determinant of A . Hence the least common multiple U ofthe denominators of the coordinates of the points { s , . . . s j } divides the determinant V of A . V is the ´A. G.HORV´ATH volume (or index) of the centering. The relevant rows of a admissible centering are the coordinates of { s , . . . s j } with respect to the basis { a , . . . , a n } . Table 3 contains the data of the admissible centeringsup to dimension 6.(3) dimension U V relevant rows2 1 1 0 ,
03 1 1 0 , ,
04 1 1 0 , , ,
04 2 2 1 / , / , / , /
25 1 1 0 , , , ,
05 2 2 1 / , / , / , / ,
05 2 2 1 / , / , / , / , /
26 1 1 0 , , , , ,
06 2 2 1 / , / , / , / , ,
06 2 2 1 / , / , / , / , / ,
06 2 2 1 / , / , / , / , / , /
26 2 4 1 / , / , / , / , , / , / , , , / , / , , / , / , / , /
26 3 3 1 / , / , / , / , / , / Characterization of the Minkowski-reduced forms by a finite system of inequalities.
We turn now to the third from the following three question of reduction theory: • How we can find a something-like reduced basis in a given lattice? • How we can find a minimum vector or a short vector in a given lattice? • How we can find all minimum vectors in a given lattice if we know one of its reduced basis?In low-dimensional cases the first two questions can be solved. The reason is that Tammela in [24]up to dimension six gave a chance to search Minkowski and (also Hermite) reduced basis. We recallhis argument. Every basis { e , . . . , e n } of a lattice determines a point in the N =: (cid:0) n +12 (cid:1) -dimensionalEuclidean space by the coefficients of its symmetric Gram matrix. The set of all such points is an openconvex cone in R N with vertex the origin. In this cone the Minkowski-reduced forms set a sub-domain isdenoted by M . A positive quadratic form Q (also corresponding to the Gram matrix of the given basis) isMinkowski-reduced if for every point u = n P i =1 u i e i ( u i ∈ Z ) with g . c . d( u i , . . . u n ) = 1 holds the inequality Q ( u ) ≥ Q ( e i ). (It can be seen easily that in this case the basis { e , . . . , e n } is a Minkowski-reduced basisof the corresponding lattice.) Now the domain M ∪ { } is a closed convex cone with apex 0. It can begiven the explicit description of M for n ≤
6. One can specify a concrete finite system of inequalities Q ( u ) ≥ Q ( e i ) that determine M . It consists of the n − Q ( e i +1 ) ≥ Q ( e i ) and inequalities Q ( u ) ≥ Q ( e i ) for which u can be transformed into a column of the matrix(4) by permuiting the coordinates of u and omitting the signes of these coordinates, while g . c . d( u i , . . . , u n ) =1. This result was announced by Minkowski for n ≤ ≤ n are taken in dimension n ) and proved by Minkowski for n ≤ n = 5 and by Tammela for n = 6. The result was slightly refined and extended to the case n = 7 by Tammela in [26].This result says that { e , . . . , e n } forms a Minkowski-reduced basis in its lattice L [ e , . . . , e n ] if for theabove finite number of inequalities Q ( u ) ≥ Q ( e i ) hold with the vectors permitted by the table. Contrary, if we find a vector u permitted by the table for which the reverse inequality hold the basis is not reducedby Minkowski. In general to find algorithmically a shortest vector (minimum vector) in a lattice is a hardtask so to find a Minkowski-reduced basis in a lattice is not easy, too. First Beyer, Roof and Williamsonin [4], and Beyer in [5]) did an algorithm for determination of a Minkowski-reduced basis. In practice,this algorithm is used only n ≤
6. The Minkowski reduction theory can be extended to integral n × m matrices with respect to the module M which is the module over the integers generated by the columnsof the given n × m matrix A . Zassenhaus and Ford indicated that the number of operations neededto reduce the matrix decreases sharply if m fixed and n tends to infinity. This fact was establishedby Donaldson in [7] using probability method. In his paper by the assumption that we need to checkonly finite number of inequalities to identify a reduced matrix, was given a sketch of an algorithm asfollows: Let Q be the quadratic form corresponding to the given basis { l . . . , l n } . We test all the finitelymany inequalities. If all are satisfied, we are done. If contrary for a row s , . . . , s n in the table we have Q ( l k ) > Q ( s l + . . . + s n ), we may apply an unimodular transformation to replace l k by s l + . . . + s n l n and leave l , ..., l k − fixed (that this is possible follows from the fact that g . c . d( s k , . . . , s n ) = 1. Eventually,all the inequalities must be satisfied, and the matrix will be Minkowski reduced.A little bit later Afflerbach and Groethe presented a new algorithm which is more practicable forgreater dimensions and requires less computation time (see in [2]). The algorithm presented in theirpaper was used to determine Minkowski-reduced lattice bases of pseudo-random number generators upto dimension 20. We suggest for the interested reader the paper of Agrell, Eriksson, Vardy and Zeger(see [3]) to a good review on the algorithmically point of view of the closest lattice point and reducedbases problems.To recall again Tammela’s work we note that his method can be used to determine the Dirichlet-Voronoi cell of a lattice point, too. Note that the Dirichlet-Voronoi cell of a lattice point contains thosepoints of the space which are closer to it as to any other lattice-point. The cell is a convex polytope, whichfacets are determined by certain lattice vectors; these vectors are their normal vectors and go throughthey middle points, respectively. We call them relevant vectors of the cell. It can be seen easily, that allminimum vectors of a lattice are relevant vectors of the Dirichlet-Voronoi cell of the origin. We can readmore to the Dirichlet-Voronoi cells in papers [9, 10, 11]. In his work [25] proved that with respect to aMinkowski-reduced basis, the integer representation of the relevant vectors of the Dirichlet-Voronoi cellof the lattice can be found among those lattice vectors which can be transformed into a column of thematrix of Table 5(5) m by permuiting the coordinates and omitting the signes of these coordinates, if we exclude the last column(1 , , , , , m ) T of the second part. The reason of the partitioning of the matrix is that the first partis Table 4 can be used to the reduction process. The second part (with the mentioned column) holdsthe property, that contains those vectors which could be shorter than the longest reduced-basis vectorneeded to its representation. In dimension n we have to take into consideration only those vectors whichhave less or equal to n non-zero coordinates.Since a vector of minimal length is always relevant vector of the Dirichlet-Voronoi cell of the origin,its coordinates with respect to a Minkowski-reduced basis can be permuted one of the column of thistable if we also omit the signs of the coefficient.5. The coordinates of a minimum vector in a Minkowski-reduced basis
Let n ≤ L [ e , . . . , e ] is spanned by a Minkowski-reduced basis. Com-bining the results on admissible centerings with the results of reduction theory we can prove a littlesharpening of Tammela’s result on the coordinates of the minimum vectors. From the Tables 5 we seethat the maximum of the absolute value of the coordinates in dimension 5 and 6 are 3 and 4, respectively.We prove the following: Theorem 1.
Let n ≤ be the dimension of a lattice L . The absolute value of a coordinate of a minimumvector with respect to Minkowski-reduced basis of L are less or equal to the maximum of the denominators ´A. G.HORV´ATH of the relevant rows in the n -dimensional Table of n -dimensional admissible centerings. Precisely, | x i | ≤ if n = 2 , if n = 4 , if n = 6 . First we highlights some easy geometric observations which we continuously use in our proof. (A)
Since we prove step by step the statement from lower to higher dimensions, without loss ofgenerality we can assume that there is no zero coordinate of the investigated minimum vector m = P x i e i . (B) If a parallelepiped Π[ a , . . . , a n ] has a k -dimensional centered face of denominator 2 then itsvolume is even. In fact, if the centered face is Π[ a , . . . , a k ] then the coordinates of the vector b =(1 / a + . . . + a k ) ∈ L [ e , . . . , e n ] are integers, respectively. Hence the volume of Π[ b, a , . . . , a n ] isalso an integer. But vol(Π[ b, a , . . . , a n ]) = det[ b, a , . . . , a n ] = (1 /
2) det[ a + . . . + a k , a , . . . , a n ] =(1 / a , a , . . . , a n ]) proves the statement. (C) Since the minimum vectors of the lattice L [ e , . . . , e n ] are relevant vectors of the Dirichlet-Voronoicell of the origin (see [9]), so by Theorem 2 in [25] the absolute value of its coordinates with respectto a Minkowski-reduced basis can be found in that modification of Table 5 in which we omittedthe last column of the middle part. From this immediately follows that for all i | x i | ≤ (D) We recall a result of G. Cs´oka from his paper [6]. It says that in a n -dimensional lattice with n ≤ L ⋆ -reduced forms are agree for n ≤ thefirst four elements of a Minkowski-reduced basis gives a primitive system of successive minimumvectors, and so Ryskov’s affinity sends these vectors to a primitive system of minimum vectors ofthe image lattice without changing the coordinates of a minimum vector. Hence we can assumethat | e | = | e | = | e | = | e | = | m | = 1 (E) Assume that n ≤ { e , . . . , e n − , x } be a primitive system of n − | e | ≤ . . . ≤ | e n − | ≤ | x | . Additionally we assume that the lattice is spanned by such Minkowski-reduced basis { e , . . . , e n } for which e n − Lin[ e , . . . , e n − , x ]. Then there is an orthogonalaffinity ϕ with ratio (1 + ε ) with positive ε , such that hold the following two properties: – It fixes the vectors of the subspace Lin[ e , . . . , e n − , x ], – the image vector e ′ n − := ϕ ( e n − ) is a shorter vector among those vectors which are not lyingin the subspace Lin[ e , . . . , e n − , x ] and completes { e , . . . , e n − } to a primitive system.Clearly, e n − ”shows to the first layer” with respect to the sublattice L [ e , . . . , e n − , x ] and it isa shorter vector in this layer. Consequently, if we decompose z to the orthogonal components z and z , the minimum value of the formula | ϕ ( z ) | = | z + (1 + ε ) z | for every positive ε attendsat e n − if we take into consideration the vectors z ∈ L [ e , . . . , e n − , e n − , e n ] \ L [ e , . . . , e n − , x ].Let ε such that the previous minimal value let be greater than the length of x . Now in this lattice { e , . . . , e n − , e ′ n − } is not a part of a Minkowski-reduced basis, hence there is a column α := α e + . . . + α n − e ′ n − + α n e ′ n in the n -dimensional part of Table 4, which exclude e ′ n − from thepossible Minkowski reduced-basis elements. But the vector corresponding to the row α has to beshorter then e ′ n − . Hence it is in the original layer implying that α = y e + . . . + y n − e n − + y n − x where y i ’s are all integers. Consequently, at the same time hold the equalities α n − = y n − x n − and α n = y n − x n . (Here α n − , α n are such non-zero integers which absolute values are in Table4.) Remark 1. If { e , . . . , e n } is a basis with | e i | = 1 for all i then the statement of Theorem 1 eas-ily proved with the equality vol(Π[ e , . . . , m, . . . e n ]) = | x i | vol(Π[ e , . . . , e n ]) . In fact, the admissiblevolumes in Table 3 are greater than 3 only in one case when n = 6 and the centering of the par-allelepiped Π[ e , . . . , e i − , m, e i +1 . . . e n ] has volume 4. However in this case the corresponding low-dimensional sublattice L ′ [ e , . . . , e i − , e i +1 . . . e n ] contains a four-dimensional centered cubic lattice L ′′ with index with respect to L ′ [ e , . . . , e i − , e i +1 . . . e n ] and thus e , . . . , e i − , m, e i +1 . . . e n ] ≥| x i | ind( L ′′ /L ′ [ e , . . . , e i − , e i +1 . . . e n ]) = 2 | x i | implying that in this case also hold the inequality | x i | ≤ .Proof of Theorem 1. Let { e , . . . , e n } be a Minkowski-reduced basis and m = P x i e i is a minimum vectorin the lattice L = [ e , . . . , e n ]. Assume that the volume of the basic parallelepiped Π[ e , . . . , e n ] is equalto 1. By (A) and (D) we can assume that the coordinates x i are non-zero and the first four elements of the basis are minimum vectors. By Remark 1 the statement is trivial in dimensions n ≤ n = 5 and n = 6, respectively. n=5 Consider the four-dimensional minimum parallelepiped Π[ m, e , e , e ]. Its edges have length 1 andso the corresponding system of vectors is a system of successive minima. Then e be a fifth successiveminima, so the lattice L [ e , . . . , e ] is an admissible centering of the lattice L [ m, e , e , e , e ]. Hencevol L [ m, e , e , e , e ] = | x | ≤ { e , e , e , m } is a primitive system then there is a Minkowski-reduced basis with first fourelement { e , e , e , m } . We use here (E) with choice x = m , so the coefficients x i are non-zero integers.Since by the 5-dimensional part of Table 4 for all i | α i | ≤
2, and since | y | ≥ | x | ≤
2, too.Assume now that { e , e , e , m } is not a primitive system in L [ e , . . . , e ]. Then L [ e , . . . , e ] ∩ Lin(e , e , e , m) is an admissible centering of L [ e , e , e , m ]. Hence the lattice L [ e , . . . , e ] ∩ Lin(e , e , e , m)is a centered cubic lattice of dimension four and volume 2. Since e is a fifth vector with minimal lengthfor which { e , e , e , e ⋆ , e } should form a basis (there is no admissible centering with volume 4) we canconclude by (D) that e is also a minimum vector. We can finish the proof of this case using the resultof the Remark 1, so in this case also hold | x | ≤ | x i | ≤ i = 1 , , i = 4 can be applied again. n=6 We use that the statement is true for n ≤
5. Let { e , . . . , e } be the Minkowski-reduced basis, and m = P i =1 x i e i is the minimal vector with non-zero integer coefficients.We have now two cases. If { e , . . . , e , m } is a primitive system then | e | = 1 and Π[ e , . . . , e , m ] is a minimum paral-lelepiped with volume | x | . By (C) | x | ≤ | x | 6 = 4. By Table 3 the onlylattice which can be centered by volume 4 is the 6-dimensional cubic lattice. The mentioned centeringcan be realized metrically only in one way (see [23] Theorem 6). Here that shortest vector which lin-early independent from { e , . . . , e } and completes it to a basis is also a minimal vector. Hence we have | e | = 1. Now we can use again the last argument of Remark 1 showing that all coordinates of m is lessor equal to 3, as we stated. If { e , . . . , e , m } is not a primitive system then the face Π[ e , . . . , e , m ] is a centered facet of theparallelepiped Π[ e , . . . , e , m, e ]. By (B) | x | = vol(Π[ e , . . . , e , m, e ]) is even. (We note that in thiscase immediately cannot apply the Table 3 because | e | > (C) it is either 2or 4. • Assume that | x | = 2. Then there is a vector e ⋆ ∈ L [ e , . . . , e , m ] for which { e , . . . , e , e ⋆ } is aprimitive system. By Table 3 this vector e ⋆ is of the form either 1 / e + . . . + e + m )+ L [ e , . . . , e ]or 1 / e + . . . + e i − + e i +1 + . . . + e + m ) + L [ e , . . . , e i − , e i +1 , . . . e ]. – In the first case, e ⋆ = P i =1 (( x i + k i ) / e i + ( x / e + e , where the coordinates areintegers. Now we can apply (E) with x := e ⋆ and so we have an integer vector α forwhich by the integer coefficient y n − hold simultaneously the equalities α = y ( x /
2) and α = y . Since | α | = | y | ≤ | α | = | y || x / | ≥ | y | = 3and | x | ≤ | y | = 2 and | x | ≤
3; or | y | = 1 and | x | ≤
6, respectively. By (C) we have to exclude only that | x | = 4 could hold. Indirectly assume that | x | = 4 andconsider the system { e , e , e , m, e } . By the integer coordinates of e ⋆ x is odd, hence e is not in the lattice Lin[ e , e , e , m, e ] ∩ L [ e , . . . , e ] and we can use again (E) . Fromthis follows that there is an integer vector β = β e + β e + β e + β e + β e ′ + β e ′ ∈ Lin[ e , e , e , m, e ] ∩ L [ e , e , e , e , e ′ , e ′ ] with coordinates | β i | ≤
3. Here m = x e + x e + x e + x e ′ + x e ′ + x e . β has the form y e + y e + y e + y m + y e where | β | = | y x | ≤ | y | = k/ k = 1 , k = 0 implies that β ∈ L [ e , e , e , e ]therefore e cannot be chosen to the sixth element of a Minkowski-reduced basis). But y x is also an integer hence x is even which is a contradiction with the fact that x is odd andby assumption it is non-zero. We finally got that in this case | x | ≤ | x | = 2because it is also non-zero and even. Investigate now the first four coordinates. By theabove argument | y | = k/ k = 1 , x is odd then k = 2, | y | = 1 and3 ≥ | β | = | x | proves the required inequality for the fourth coordinate x , too. Clearly, the ´A. G.HORV´ATH first four coordinates have equivalent role in our arguments, so for all i in this case hold theequality | x i | ≤ – Secondly, we assume that e ⋆ = P i =2 (( x i + k i ) / e i +( x / e + e , hence e ⋆ gives a admissiblecentering of the minimal 4-dimensional parallelepiped Π[ e , e , e , m ]. This means that e ⋆ isa minimal vector, and { e , e , e , e , e ⋆ } is a primitive system of minimal vectors, implyingthat e is also minimal vector. Since { e , e , e , e , e , e ⋆ } is also a basis then e is a minimalvector and we can apply Remark 1. Hence the absolute values of the coordinates of m areless or equal to 3. • To end the discussion, assume that { e , . . . , e , m } is not a primitive system and | x | = 4. Thenthe parallelepiped Π[ e , . . . , e , m, e ] has volume 4. On the other hand let a be a sixth vec-tor which lengths is the sixth successive minimum of the lattice with respect to the system { e , . . . , e , m } of minimum vectors. The volume of the parallelepiped Π[ e , . . . , e , m, a ] is lessor equal to 4, because its admissible centering is the lattice L [ e , . . . , e ]. – If vol(Π[ e , . . . , e , m, a ]) = 2 and e ⋆ is centering the facet Π[ e , . . . , e , m ] then { e , . . . , e , a , e ⋆ } is a basis. If | e ⋆ | ≤ | a | then | a | ≥ | e ⋆ | ≥ | e | implying that | a | = | e | . Now { e , . . . , e , m, e } a system of successive minimum vectors and so the centering of Π[ e , . . . , e , m, e ] is thesame as the admissible centering of the cubic lattice with index 4. But in this lattice thereis no minimum vector with coordinate 4. This is a contradiction. So we have | e ⋆ | > | a | and so { e , . . . , e , a , e ⋆ } is an ordered basis. Hence again | a | = | e | leads to the samecontradiction as above. – If vol(Π[ e , . . . , e , m, a ]) = 4 then the centering is combinatorially agree with the centeringof the 6-dimensional cube with index 4. Hence there is a 4-dimensional centered face byindex 2 which contains a as an edge vector. Denotes by x the shortest half diagonal ofthis face. Its length is smaller than the half-diagonal of a brick with the same edge lengthsand greater than the length of a . Thus | a | ≤ | x | ≤ / | a | ) so | a | = 1 and thesuccessive minimum vector is in fact a minimum vector. The lattice L [ e , . . . , e , m, a ] is the6-dimensional cubic lattice and L [ e , . . . , e , e , e ] is its admissible centering. In this latticethe Minkowski reduced basis contains six minimal vectors and so we can apply Remark 1,again. ✷ References [1] Afflerbach, L.: Minkowskische Reduktionsbedingungen f¨ur positiv definite quadratische Formen in 5 Variablen
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Math. Comput. (1971) 345–360.[5] Beyer, W. A. : Lattice structure and reduced bases of random vectors generated by linear recurrences. In: Applicationsof Number Theory to Numerical Analysis (Zaremba, S. K., ed.) , (1972) 361–370.[6] Cs´oka, G.: On an extrem property of bases reduced by Minkowski. (in Russian)
Studia Sci. Math. (1978) 469–475.[7] Donaldson, J. L.: Minkowski Reduction of Integral Matrices MATHEMATICS OF COMPUTATION , (1979)201–216.[8] G.Horv´ath, ´A.: On n -dimensional lattice bases reduced by Minkowski and by Hermite (in Hungarian) Mat. Lapok (1982-1986), 93–98.[9] G.Horv´ath, ´A.: On Dirichlet-Voronoi cell. { Part I. Classical problems } Per. Poly. ser Mech. Eng. (1995), 25–42.[10] G.Horv´ath, ´A.: On the Dirichlet-Voronoi cells of unimodular lattices.
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