UvA-DARE (Digital Academic Repository) Lattices, codes and Radon transforms Boguslavsky, M. Link to publication

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1 UvA-DARE (Digital Academic Repository) Lattices, codes and Radon transforms Boguslavsky, M. Link to publication Citation for published version (APA): Boguslavsky, M. I. (1999). Lattices, codes and Radon transforms General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 15 Nov 2018

2 Chapter 2 Homogeneous spaces in duality In this chapter, we mention first some notions from the theory of homogeneous spaces in duality as developed in [Hell] and [Hel2]. This theory delivers a unified point of view on many dualities arising in coding and lattices theories. We prove relations on T-functions, two known bounds on generalized Hamming weights and generalized Hermite parameters, and give a new interpretation of Nogin weight/multiplicity duality. It seems to us that many other applications of these technique are possible. For example, in proofs of Theorems 4.1, 4.2, and 4.3 in Chapter 4, the key step is equivalent to the use of the Plancherel formula for a suitable Radon transform in P m (F 9 ). 2.1 General Theory In this section, we follow the books [Hell] and [Hel2]. Let G be a locally compact group, X and H two (left) coset spaces X = G/Hx and H = G/H^, where Hx and H^ are two closed subgroups of G. Let K be the intersection IflS. Let us make the following assumptions: (i) The groups G, Hx, H=, Hx n H-= are unimodular (i.e. the leftinvariant Haar measures are right-invariant); (ii) For any hx Hx the inclusion hxh-= C H-^Hx implies hx H-=; 21

3 Homogeneous spaces in duality for any h E G H E the inclusion h E H x C H X H E implies h E G H x \ (iii) The set H X H E is closed. Homogeneous spaces X and E are called homogeneous spaces in duality. We shall say that x G X and G E are incident and denote it by x tx if the cosets xh x and H E are not disjoint. The classical example of the homogenous spaces in duality is the pair (points in R", hyperplanes in R n ) with the incidence relation (x M Other examples are the pair of real Grassmanians (Ç(n, m),ç(n,nm 1)), ([Hell]), symmetric spaces, complex spaces, and quadrics in C ([GGV]), The transform considered in 2.3 can be regarded as the F 5 -analogue of the real X-ray transform, widely applied in radiology and tomography [LS]. We put x = { G E : x M } C E, f = {x G X : M x} C X. The factor G/K may be identified with the set {(x, ) G X x E : x tx f}. The maps x i-)- x and H>- can be also described via the double filtration G/K P / \ *, (2.1) X = G/H X E = G/H E where p(## x D ff s ) = gh x and TT(## X n # s ) = gh s. Namely, x = 7r(p-\x)),i = p(n- 1 (0). Given Haar measures that satisfy (i) we may construct nice G- invariant measures m(x) on each f and /i(f) on each x (cf. [Hell, p. 143].) by The Radon transform ƒ : E -> C of a function ƒ : X -> C is defined /(0= fj(x)dm(x); (2.2) the duaj Radon transform <j) : X» C of a function </> : H > C is defined by 0(z) = I m dm- (2.3) 22

4 CHAPTER 2 Lemma 2.1 flhell], Plancherel formula.) Let ƒ : X -> C and </> : E» C 6e continuous compact support functions. Then ƒ and f are continuous and J x f(x)4>(x) dx = j[ ƒ (0^(0 de- (2.4) Note. For a discrete group G the formal equalities Y,mk*)= /(*Mö = /«Mö (2.5) show that Eq. (2.4) holds also for any functions ƒ and < such that all series in (2.5) converge absolutely. The proof for the general case is similar but requires some additional facts about measures and groups. Actually, equality (2.4) holds in a more general case of a double filtration like (2.1) than that of conditions (i)-(iii). However, the existence of a nice group structure is often useful and helps to choose the right homogeneous space representation. The problem of the inversion of a Radon transform (2.2) and of the dual transform (2.3), is, in general, rather complicated 1, and there is no general inversion formula. In some special cases this problem was solved. For the classical case of the pair (R n, hyperplanes in R n ) this problem was solved by J. Radon [Rad]. The inversion formulas are quite different in the cases of the even and odd dimensions. For a pair of real Grassmannians, the Radon transform was inverted by Helgason [Hell]. We did not find an inversion formula for lattice spaces being investigated in the next section. The case of codes is simpler and an inversion formula for a Radon transform in a projective space over a finite field is obtained in Section 2.3. An equivalent result was proved by Nogin [Nog3] in connection with one problem about projective multisets. : A Radon transform often maps some "non-essential" functions to the all-zero function (see [GGV].) In the case of a vector space over a finite field these "nonessential" functions are somewhat similar to latin squares (see the cover.) Indeed, take the rows and the columns of a latin square as one space H, and the cells as the other space X. The numbers in cells define a certain function on X and the Radon tranform of this function is constant on E. 23

5 Homogeneous spaces in duality 2.2 A Duality Between Vectors and (n 1)- sublattices Let A(n) denote the group of integer n x n matrices with the determinant ±1: A(n) := {M G GL n {Z) : det M\ = 1} ~ SL n (Z) x {±1}. The subset G = R(n) C GL n+1 (Z) defined by G - *( ) := ( ^ * ) (2.6, is a group with the respect to the usual matrix multiplication. The map x i-)- Mx = M'x + v, M I ) G R(n),x e Z n defines an action of i?(n) on Z n. Note that this is also a transitive action of R(n) on the set of all shifts of bases of Z n. Let Hx denote the stabilizer of the point 0: H x = St(0) ~ A(n); (2.7) let n be a shift of an (n l)-sublattice of Z n and let i/n denote the stabilizer of IT. Assume now that II is the sublattice Ilo C Z" spanned by the first n 1 base vectors; then H s = St(U 0 ) = St(v u v 2,...,» _!> ~ Ä(n - 1) x {±1}. (2.8) Lemma 2.2 The spaces X = G/Hx and S = GjH-= defined by Eq. (2.6), (2.7) and (2.8) satisfy conditions (i), (ii) and (Hi). PROOF. Conditions (i) and (iii) are obvious. Let us check condition (ii). We need to prove that if hx G Hx is such that for any h-= G H-= there exist gx G Hx and g-= G H-= satisfying h x h s = g~g x (2.9) then hx G Hs- Applying the right and the left hand sides of (2.9) to the origin, we get hx(h s 0)=g a (g x 0). (2.10) 24

6 CHAPTER 2 Since g x 0 = 0, the right hand side of (2.10) belongs to n 0. It is clear that H-= acts transitively on n 0, so /i=0 runs through Ilo as /i= runs through H E. Thus, h x (p) G n 0 for any p 6 n 0) i.e. h x G H3. The dual statement of (ii) is proved similarly. A Recall that by L^m' we denote the set of all primitive m-sublattices of a lattice L and denote by LH the set of all shifts of primitive m- sublattices of L by vectors of L. The intersection of H x with Lf s is A(n-l) 0 0 K:=H x nh E = 0 ±10 I ~A{n-l) x {±1} The factorspace X = G/H x can be identified with Z n, and the factorspace S = G/H^ can be identified with the set (Z")^n_1 ' of all integer shifts of primitive (n l)-sublattices in Z. It can be checked that with our choice of Ilo a point x G X is incident with a shift of sublattice e G s iff x G e Note. A different choice of Ilo m (2-8) will give a different incidence relation; for example, when H-= stabilizes n = n A = (vi, u 2,..., u n _i) + \v n, A G z, we get the incidence relation x tx 44> ind^^a;, ) = A. The Plancherel formula (2.4) gives ƒ(*)*(*) = E /(0^(0 (2.11) xez" fe(z") ln - 1] for any ƒ : L > C and </> : L^" -1 '» C such that both series converge absolutely. Theorem 2.1 For any lattice L of rank n Q L (q) T - l (p) = E_p ll$li e f (g). (2.12) 25

7 Homogeneous spaces in duality PROOF. Let us apply Eq. (2.11) to f(x) = gll x 1 and 0(f) = pikh, where the norms are given by the positive quadratic form associated to L. By the definitions, Q L {q)-tr\p) E i M E p" = E xez" / \çe(z") 1 7 çiixii j2 p m 7 Shift a sublattice f e Ü- n ^ by a vector x. We can replace now a summation over all f G L'" -1! by a summation over all f x = f + x G Li"" 1 ], ll^ll = f and apply then the Plancherel formula (2.11): E (V"EH= W \ S P KI (2.11) gimi p * ^ çlmiplieil = ^p^e,(g). A Thus, we proved that the product of the 0-function of a lattice with the T n_1 -function equals the weighted sum of shifted sublattice O-functions. Applying the duality relations (1.7), (1.14) we get the following corollary Corollary 2.1 For any lattice L CÄ holds e L (<i)-tl(p detl )= E p m d<i)- (2.13) 2.3 Weight/Multiplicity Duality for Projective Multisets Nogin [Nog3] proposed the following construction of new linear codes from the known ones. 26

8 CHAPTER 2 The projective multiset Yc of a linear [n, k, d] q -code C (see 1.1.2) can be considered as a multiset of n hyperplanes with multiplicities v(h) in the projectivization PC of the code C. Assign multiplicity zero to any hyperplane not in the multiset. The weight of a 1-subcode c G PC equals wt(c) = 2>(JÏ). (2.14) A natural problem is to invert relations (2.14), i.e. given the set of weights {wt(c) c G PC} one wants to reconstruct the multiplicities {u(h)}. Nogin proved the following inversion formula for (2.14): u{h) = ^Pcwt(c)-gE c g wt(^ (2 i5) Now, for any given function wt : P _1» Z one can reconstruct a set of "multiplicities". These "multiplicities" do not necessarily correspond to an actual set of multiplicities of a projective multiset. However, they can be corrected to an actual set of multiplicities by a linear transform (see [Nog3]). Nogin used this inversion to construct new long linear codes: one can take a "small" code Ci, construct from it in a certain way a set of "weights", apply Eq. (2.15) to obtain a set of "multiplicities" and correct them to a set of multiplicities of a projective multiset. The code Ci corresponding to this multiset is much longer than C\ and its spectrum is determined by the spectrum of C\. This construction has a natural interpretation via a Radon transform. Consider the group G := PGL(k l, q ) with the standard action on PC and subgroups H x := St(P) and H-= := St(H), where P is an arbitrary but fixed point in PC, and H is a hyperplane containing P. The conditions (i)-(iii) (see page 21) can be easily checked, so the pair (X = G/Hx,z, = G/H-=) is a pair of homogeneous spaces in duality. The incidence relation is the Radon transform and the dual are given by HO = /(*)> (2-16) m = i>(0- (2-17) Ç3x 27

9 Homogeneous spaces in duality These transform can be inverted in the following way. Consider a function ƒ : X -> R. We want to express ƒ via its Radon transform ƒ. Let p m denote the number of points in an m-dimensional projective space over q, p m = ^^1" 1. Let us introduce the following functionals s ((f)), <j(f) and operators D(f>, A/ defined on the function spaces {(f) : S -» C} {ƒ : X -> C} by Pra-l Pm-1 Theorem ie Radon transform (2.16) and the dual Radon transform (2.17) are inverted by the formulas PROOF. It is sufficient to prove Eq. (2.18) for the indicator function of the one-point set {P}, i.e. for the function Ip W = 1 0, x? P The result can be then extended to arbitrary functions by the linearity of the Radon transform. For Ip(x) it is clear that /p(0 = { o', UP ; S(lp) = Pm-l SO A Note that Eq. (2.14) can be rewritten as wt(c) = n v(c). Using the inversion formula (2.18) for the function v(c) we get Eq. (2.15). 28

10 CHAPTER Generalized Plotkin Bound In this section, we prove a lower bound for generalized Hamming weights (see [TV2] and [HKYL].) We show that this bound can be regarded as a corollary of the Plancherel formula. We use the following lemma, which is due to van der Geer and van der Vlugt: Lemma 2.3 ([GVl]) For any r-subcode D holds wt( >) = i r wt(c). y y c D (2.19) Let the incidence relation between r-subcodes of a code C be given by ctxi D & c G D, where c 6 C is a codeword and D C C is an r-subcode. Theorem 2.3 For any linear [n, k, d] q -code C and f or any r = the following holds l,...,k E wt(d) T3I PROOF. We use twice Eq. (2.19) (lemma 2.3) and once Eq. (2.4) (lemma 2.1): r L fc J q E DeCM w t(d) (2.19),. (2.4) 1 - S Ew.(c) l =' DeCH CM ) - T E #{D e ch c e D} wt(c) = 1 <7 r r-1 -Ç (2.19) 1 q T r-l -q nq k ~ T q k - 1 r L j q (2.19) wt(c)^ J 5 cec 29 (g fc -g fc - x )wt(c) y q cec Ocxc

11 Homogeneous spaces in duality A An easy corollary of this theorem is the following bound on d r (this is Theorem 1.1 from [TV2].) Corollary 2.2 ([TV2]) The r-h generalized Hamming weight d r (C) of an [n, k, d] q -code C satisfies A tr\ < n (g r -!)g fc " r d^c> ^ ^ittl ' 2.5 Generalized Minkowski Hlawka Theorem This theorem is a nonconstructive lower bound on 7 >m. It was first proved by Thunder [Thul] in greater generality (see ) We give a shorter and simpler proof of this theorem based on the Plancherel formula. Let Z(j) = C0)r(i/2)/7T J/ ' 2, where Ç(j) is the Riemann ^-function Theorem 2.4 ( Generalized Minkowski-Hlawka Theorem) For any m < n there exists a lattice L C M" with j m (L) > (n U^n- m+ \ Z^). (2.20) For m = 1 we get the classical Minkowski-Hlawka theorem. The idea of our proof is the same as in Siegel's proof of Minkowsi-Hlawka theorem [Sie]. The main ingredient of that proof is Siegel mean value theorem, which states that for a compactly supported continous function <j> : M. n -> R, f é(x)dx = an)f Y,<t>{gz)dg, (2.21) where C n is the factorspace S, L n (R)/5L n (Z) with the Haar measure dg scaled so that vol(5l n (R)/5L n (Z)) = 1 and P is the set of primitive 30

12 CHAPTER 2 integer vectors in R. We shall prove a generalization of this mean value theorem is a corollary of the Plancherel formula for the pair of homogenous spaces in duality (C n,tz m ), where 1Z m is the space of all m-lattices in R n, and the incidence relation is [xi M < > "M is a primitive sublattice of L". (2.22) The standard way to represent C n as a homogenous space is to identify it with S L n (R) / S L n (Z). However, it is more convenient for us to identify it with the factorspace of real unitary matrices by the integer unitary matrices. Let U n (R) ~ SL n (R) x {±1} be the subgroup of GL n (R) consisting of all matrices A with deta = 1 and let U n (Z) ~ SL n (Z) x {±1} be the subgroup of all integer matrices in [/ (E). It is clear that C n = U n (R)/U n (Z) (2.23) and that U n (M) acts transitively on the set 7Z m. Thus, n m = U n (R)/St(M 0 ), (2.24) where M 0 is any fixed m-lattice in E". In coordinates, when M 0 is spanned by the first m basis vectors we have In the sequel, the integrations over C n and TZ m are assumed to be with the respect to the measures induced by the Haar measure on f/ (M) scaled so that vol n = 1. One can check that the pair of spaces ( n,lz m ) is a pair of homogenous spaces of duality in the sense of conditions (i)-(iii) on page 21with the incidence relation (2.22). Theorem 2.5 (Generalized Siegel mean value theorem) Let < ( ) be a compactly supported function on TZ m. Then where c I Y. ^( M ) dl = f ^( M ) dm i ( 2-25 ) MxiL T(n/2 + l) V T\7=2 Z(J) J 31

13 Homogeneous spaces in duality- Note. For m = 1 we have C = 2((n) and not ((n) as in in (2.21) because our space 7?-i does not coincide with R n but is in fact the factor R n /{±1}. PROOF. We should simply use the Plancherel formula (2.4) for the pair (jc n,lz m ) of homogenous spaces in duality defined by (2.23) and (2.24). It is easy to check that M e 1Z m is incident to L E C n iff M is a primitive m-sublattice of L. Take <f>(-) as the function on 1Z m and f(l) = 1 as the function on C n. We have / f(l)$(l)dl= f <j){m)hm)dm. JC n Jllm Substituting ƒ (L) = 1 and using the definitions we get ƒ J2 4>{M) dl= f vol(m)(/.(m) dm. It is clear that vol(m) is independent of M. In fact, it equals the volume of the space of all n-lattices with a fixed m-sublattice. Define the constant C by 1/C = vol(m). Thus C I Y, ^( M ) dl = i <M M ) dm. Similarly to Siegel's argument, one computes now the value of C via a rather technical inductive calculation in the space of matrices. A Theorem 2.4 follows now by the standard argument: let B R C lz m be the ball B R = {M\ det M < R 2 } of radius R and let <j>(m) be the indicator function of this ball. The right hand side of Eq. (2.25) equals \o\{b R ). So if R is such that vol B R < C, then EMCXL 4>(M) < 1 for at least one L G C n. Thus, any m-sublattice of L is outside B R, so lm(l) > R 2. This completes the proof of Theorem 2.4. A The bound of this theorem is asymptotically good, that is, it differs from the lower bound (1.19) just by a multiplicative constant. 32

14 CHAPTER 2 Note that a similar bound for generalized Hamming weights of codes (generalized Gilbert-Varshamov bound, see [TV2]) can be proved similarly, by using the Placherel formula for the pair of homogenous spaces in duality ([n, k] q codes, [n, r] q codes). 33

15 Homogeneous spaces in duality 34