On Conjugacy Classes of Subgroups of the General Linear Group and Cyclic Orbit Codes

Home , Cyclic group

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0 1 0 ··· 0 ϕ : S → GL (F )   A n q 001 0 Ai 7→ LAiL−1  . .. .  Fs×s Mp :=  . . .  ∈ q .   is an isomorphism if restricted to the image of ϕ. As  000 1    a consequence, the generator A of SA is mapped to −p0 −p1 −p2 ··· −ps−1 −1 a generator of LSAL = SB, i.e., an element of i S The following theorem states the existence and uniqueness {B | gcd(i, | B|)=1}. Then, there exists an i ∈ N S −1 i of a rational canonical form. with gcd(i, | B|)=1 such that LAL = B , which implies that A ∼ Bi. Theorem 4 ([7, Chapter 6.7]): Let A ∈ GLn(Fq). Then c ⇐ From the hypothesis we know that hBii = S there exists a matrix L ∈ GLn(Fq) such that B and that there exists L ∈ GLn(Fq) such that A = −1 −1 i L AL = diag(M e11 ,...,M e1r1 , L B L. The statement follows as a consequence. p1 p1

e 1 e ...,Mp m ,...,Mp mrm ) (1) m m We introduce the following definition. is a block diagonal matrix where pi ∈ Fq[x] are irreducible Definition 8 ([8, Definition 3.2]): Let p ∈ Fq[x] be a N N polynomials, eij ∈ are such that ei1 ≥ ··· ≥ eiri , nonzero polynomial. If p(0) 6=0, then the least integer e ∈ eij ei1 e χA = p and µA = p represent respectively the such that p divides x − 1 is called the order of p. Qi,j i Qi i characteristic and the minimal polynomials of A and M eij The definition is generalizable to any p ∈ Fq[x] but it is not pi denotes the companion matrix of the polynomial peij . More- interesting for the purpose of this paper since we will only over, the matrix (1) is unique for any choice of A ∈ GLn(Fq). consider irreducible polynomials. Definition 5: Let A ∈ GLn(Fq). The matrix (1) is In order to give unique representatives for the classes of called rational canonical form of A and the polynomials cyclic groups contained in G/ ∼c we need the following e1r e e11 1 em1 mrm F p1 ,...,p1 ,...,pm ,...,pm ∈ q[x] are its elemen- lemma. F eA,1 eA,m F tary divisors. Lemma 9: Let A ∈ GLn( q), pA,1 ,...,pA,m ∈ q[x] its The following lemma motivates why rational canonical elementary divisors, where pA,j for j ∈ {1,...,m} are not forms are a good choice of representatives for the classes of necessarily distinct, and SA < GLn(Fq) the cyclic group GLn(Fq)/ ∼c. generated by A. Then, for every i ∈ N with gcd(i, |SA|)=1, Lemma 6: Let F . Then the following state- the elementary divisors of Ai are exactly m many. If we denote A, B ∈ GLn( q) e e Ai,1 Ai,m F ments are equivalent: them by pAi,1 ,...,pAi,m ∈ q[x], then, up to reordering, the order of pA,j is the same as the one of p i and eA,j = e i 1) A ∼c B, and A ,j A ,j 2) A and B have the same rational canonical form. for j =1,...,m. Proof: First we prove the case where the elementary This lemma is well-known and is a direct consequence of divisor is unique. At the end of the proof we will give the the uniqueness of the rational canonical form. main remark that implies the generalized statement. Now we want to extend the previous characterization to eA F Let pA ∈ q[x] be the elementary divisor of a matrix subgroups of GLn(Fq). A ∈ GLn(Fq) and k := n/eA. Let Fqk := Fq[x]/(pA) Consider the set of all subgroups of GLn(Fq) be the splitting field of the polynomial pA and µ ∈ Fqk a primitive element of it. There exists a j ∈ N such that G := {S | S < GLn(Fq)} u k−1 jq eA pA = (x − µ ). Since p is the unique elementary Qu=0 A and the following equivalence relation on it. Given S1, S2 ∈ divisor of the matrix A, it corresponds to the characteristic and G then the minimal polynomial of A. As a consequence we obtain that the Jordan normal form of A over Fqk is −1 S1 ∼c S2 ⇐⇒ ∃L ∈ GLn(Fq): S1 = L S2L. ea ea JA = diag J j ,...,J k−1 The following theorem extends the arguments of Lemma 6 A,µ A,µjq to the case of cyclic subgroups. ea F where JA,µjqu ∈ GLeA ( qk ) is a unique Jordan block with F S S u Theorem 7: Let A, B ∈ GLn( q) and A = hAi, B = diagonal entries µjq for u =0,...,k − 1. F hBi < GLn( q) be the two cyclic groups generated by them. By the Jordan normal form of A it follows that for every Then, SA ∼c SB if and only if |SA| = |SB| and there exists i k−1 i ∈ N the characteristic polynomial of A is pAi = ( x− i u u=0 an i ∈ N with gcd(i, |S |)=1 such that A ∼ B . ijq eA Q B c µ ) . Let us now focus on the i’s such that gcd(i, |SA|)= Proof: i 1. A is then a generator of SA, i.e., pAi ∈ Fq[x] is a monic ⇒ Since SA ∼c SB, it follows that there exists an irreducible polynomial whose order is the same as the one of −1 L ∈ GLn(Fq) such that SA = L SBL, implying pA. eA In order to conclude that pAi is the elementary divisor j =1,...,r. It follows that the elementary divisors i i i of A we consider its rational canonical form. Assume that of B and the ones of A are the same, i.e., A ∼c B . the elementary divisors of Ai were more than one. Without eA,1 loss of generality we can consider them to be two, i.e., pAi The theorem states that we can uniquely represent the eA,2 and pAi . This means that its rational canonical form is classes of cyclic subgroups in G/ ∼c by considering the i e 1 e 2 RCF(A ) = diag(Mp A, ,Mp A, ) where we use the operator cyclic subgroups generated by a rational canonical form based Ai Ai RCF as an abbreviation for rational canonical form and on the choice of a sequence of polynomials of the type e1 em F eA = eA,1 + eA,2. For any j ∈ N we obtain that the matrix p1 ,...,pm ∈ q[x] where the polynomials p1,...,pm are i j m RCF((RCF(A )) ) is a block diagonal matrix with at least irreducible and ej ·deg(pj )= n. Moreover, what matters Pj=1 two blocks. Let j ∈ N such that ij ≡ 1 (mod |SA|) and in the choice of the polynomials pj ’s is only their degrees and i −1 i L ∈ GLn(Fq) be a matrix such that RCF(A ) = L A L, orders. then Trivially, the following holds for the cardinality of a cyclic i j −1 i j −1 group. (RCF(A )) = (L A L) = L AL ∼c A Corollary 11: Let SA = hAi < GLn(Fq). Then the order implying that of SA is the least common multiple of the orders of the e1 em F i j elementary divisors p1 ,...,pm ∈ q[x] of the matrix A. RCF(A) = RCF((RCF(A )) ) To conclude the section we are going to give an example

This leads to a contradiction since RCF(A)= M eA has only explaining why a straight forward generalization of Theorem pA eA 10 to any subgroup of GLn(Fq) does not work. one block. We conclude that pAi is the elementary divisor of Ai. Example 12: The only difference in the case where m > 1 consists in 1) Consider the following matrix over F2: the choice of the splitting field. Given eA,1 eA,m pA,1 ,...,pA,m ∈ 0 1 0 F [x] the elementary divisors of A and p 1 ,...p with   q A,l A,lr A = 0 0 1 . l ,...l ∈{1,...,m} the maximal choice distinct polynomi- 1 r 1 1 0 als from the elementary divisors, the splitting field on which r Although the elementary divisor of and the one of the proof is based is Fq[x]/( pA,l ). A t=1 t t S We are now ready to characterizeQ cyclic subgroups of its transpose A is the same, the groups A = hAi = and F t are not conjugate. GLn(Fq) via the equivalence relation ∼c based only on their hA, Ai GL3( 2)= hA, A i F F 2 F elementary divisors. 2) Let 4 = 2[x]/(x + x + 1) and µ ∈ 4 a primitive element. Consider the following matrices over F : Theorem 10: Let A, B ∈ GLn(Fq) and SA, SB ∈ G the 4 S S cyclic subgroups generated by them. Then, A ∼c B if and 0 1 0 0 1 0     only if the following conditions hold: A = 0 0 1 , B1 = 0 0 1 , 1) A and B have the same number of elementary divisors, 1 1 0 1 0 1 and µ +1 1 µ eA,1 eA,m eB,1 eB,m 2) if p ,...,p ∈ Fq[x] and p ,...,p ∈   A,1 A,m B,1 B,m and B2 = µ µµ +1 . F are the elementary divisors of respectively and q[x] A  0 1 0  B, then, up to a reordering argument, the orders of pA,j Although B ∼c B , i.e., they have the same unique and pB,j are the same and eA,j = eB,j for j =1,...,m. 1 2 elementary divisor, it holds that , Proof: |hA, B1i| 6= |hA, B2i| meaning that the two groups are not conjugate. ⇒ By Theorem 7, there exists a power i ∈ N with i gcd(i, |SA|)=1 such that A ∼c B , i.e., they have II. CONJUGATE GROUPS AND CYCLIC ORBIT CODES the same elementary divisors. The statement follows We now apply the results from the previous section to the with Lemma 9. characterization of cyclic codes. F ⇐ Let pB,l1 ,...pB,lr ∈ q[x] with l1,...lr ∈ Definition 13: Let S1, S2 < GLn(Fq) and C1 := {U1A | {1,...,m} be the maximal choice of pairwise co- S S A ∈ 1}, C2 := {U2A | A ∈ 2} ⊆ GFq (k,n) be two orbit prime polynomials from the elementary divisors of codes. We say that C1 and C2 are conjugate or simply C1 ∼c C2 F r F B, the splitting field of t=1 pB,lt and µ ∈ if there exists a matrix L ∈ GLn(Fq) such that a primitive element of it.Q Consider the notation −1 U2 = U1L and S2 = L S1L, kj := deg pB,lj for j = 1,...,r. Then, there exist k N j −1 −1 iB,1,...,iB,r ∈ such that pB,lj = u=0 (x − i.e., C2 = {U1AL | A ∈ S1} = {U1L(L AL) | A ∈ S1}. i qu Q µ B,j ) for j = 1,...,r. The same holds for the In order to further study properties of orbit codes, we need matrix A, i.e., there exist iA,1,...,iA,r ∈ N such to introduce the notion of distance distribution for orbit codes. u kj −1 iA,j q that pA,lj = u=0 (x − µ ) for j = 1,...,r. Due to [6], we are able to adapt the definition of weight By the conditionQ on the orders, there exists a unique enumerator from classical coding theory to orbit codes. But N i ∈ such that iA,j ≡ i · iB,j (mod ord(pB,lj )) for first we recall some facts from [6]. ¯ Definition 14 ([6, Definition 3]): Let U ∈ GFq (k,n). Then U1 the stabilizer group of U is defined as U¯2 Stab(U) := {A ∈ GLn(Fq) |UA = U} < GLn(Fq). U¯3 ... The following proposition is important in order to define the distance distribution. Proposition 15 ([6, Proposition 8]): Let C = {UA | A ∈ 0 0 0 ¯ S < GLn(Fq)} be an orbit code. Then it holds that Ut |S| |C| = |S ∩ Stab(U)| d1 d2 d3 dt and Fig. 1. The matrix U in row reduced echelon form. d(C) = min d(U, UA). A∈S\Stab(U)

Deﬁnition 16: Let C = {UA | A ∈ S < GL (F )} ⊆ i j n q If N and ¯ e C := {UM | i ∈ } Ci := {rowsp(Ui)Mp i | j ∈ F i G q (k,n) be an orbit code. The distance distribution of C is N k+1 }, then the tuple (D0,...,Dk) ∈ N such that t |{A ∈ S | d(U, UA)=2i}| j ¯ ¯ e (2) Di := . d(C)≥2k−2 max dim rowsp Ui ∩rowsp UiMp i , S X j∈N i | ∩ Stab(U)| i=1 k As a consequence we obtain that D0 =1 and i=0 Di = and |C| := lcm(|Ci|,..., |Ct|). |C|. We are able to state the following theorem thatP charac- Proof: Consider the following projections terizes conjugate orbit codes and that is a generalization of Fn Fdi Theorem 9 from [9]. πi : q −→ q (v1,...,vn) 7−→ (vl −1+1,...,vl ) Theorem 17: The binary relation ∼c on orbit codes is an i i equivalence relation. Moreover, let C1, C2 be two orbit codes i where li = j=1 di for i = 1,...,t. Since (U1,...,Ut) has such that C1 ∼c C2, then |C1| = |C2| and they have the same full rank andP is in row reduced echelon form, the matrices distance distribution. ¯ ¯ Fn Ui have full rank. Let Ui ⊂ q be the space spanned by the Proof: The fact that ∼c is an equivalence relation on orbit rows of (U ,...,Ut) indexed by the rows corresponding to codes is a consequence of Theorem 7. 1 U¯i. Since U¯i has full rank it follows that πi| ¯ is injective Let S F and F Ui C1 := {UA | A ∈ < GLn( q)} L ∈ GLn( q) for i = 1,...,t. As a consequence we obtain that for any such that C2 = {UAL | A ∈ S}. The same cardinality is i =1,...,t, if we deﬁne mi ∈ N such that consequence of the fact that given A, B ∈ S then ¯ ¯ mi ¯ ¯ j N dim Ui ∩ UiM ei ≥ dim Ui ∩ UiM ei , ∀j ∈ UAL = UBL ⇐⇒ UA = UB. pi pi

mi and ¯ ¯ e , then The same distance distribution follows from the distance Vi := Ui ∩ UiMp i F i preserving property of the GLn( q) action on GFq (k,n), i.e., ¯ ¯ mi d(UL, UAL)= d(U, UA). πi(Vi) ⊆ rowsp(Ui) ∩ rowsp(UiM ei ). pi The importance of this last theorem is that two conjugate It follows that orbit codes are not distinguishable from the point of view of cardinality and distance distribution. Theorem 10 translates as ¯ ¯ j dim(Vi) ≤ max dim(rowsp(Ui) ∩ rowsp(UiMpei )). follows in the language of orbit codes. j∈N i Corollary 18: Every cyclic orbit code is conjugate to a t ¯ Since U = ⊕i=1Ui we conclude that cyclic orbit code defined by a cyclic group generated by a matrix in rational canonical form. d(C)=2k − 2max dim(U∩UM j ) j∈N This fact gives us the opportunity to consider only cyclic t orbit codes out of matrices in rational canonical form for the j ¯ ¯ e ≥ 2k − 2 max dim rowsp Ui ∩ rowsp UiMp i study of codes with good parameters. X j∈N i i=1 We are now interested in these orbits codes. The cardinality of C is a direct consequence of the fact that Theorem 19: Let M := diag(M e1 ,...,M et ) ∈ p1 pt GLn(Fq) a matrix such that pi ∈ Fq[x] are monic irreducible i i i diag(M e1 ,...,M et ) = diag(M e1 ,...,M et ) ei p1 pt p1 pt polynomials and di := deg(pi ) for i = 1,...,t. Let Fk×di U = rowsp(U1,...,Ut) ∈ GFq (k,n) with Ui ∈ q and and of the minimality of the least common multiple. where (U1,...,Ut) is in row reduced echelon form. For any It is possible to find examples for which the lower bound i ∈ {1,...,t}, let U¯i be a submatrix of Ui as depicted in given by (2) is attained. The following lemmas depict these Figure 1. examples. Lemma 20: Let M := diag(M e1 ,...,M et ) ∈ GLn(Fq) p1 pt a matrix such that pi ∈ Fq[x] are monic irreducible polyno- m g ei ¯ ¯ j mials and di := deg(pi ) for i = 1,...,t. Let k ≤ di for =2k − 2 dim(rowsp(Uj ) ∩ rowsp(Uj M ej )). X pj j=1 i = 1,...,t and U := rowsp(U1,...,Ut) ∈ GFq (k,n) where Fk×di Ui ∈ q are matrices having full rank for i = 1,...,t. If i N j we define C := {UM | i ∈ } and Ci := {rowsp(Ui)M ei | A matrix M ∈ GLn(Fq) is called completely reducible if its pi j ∈ N} and it holds gcd(|Ci|, |Cj |)=1 for all i 6= j, then elementary divisors are all irreducible, i.e., from Definition 5 if ei,j =1 for all i, j. One can use the theory of irreducible cyclic d(C) = min d(Ci). i∈{1,...,t} orbit codes from [9] to compute the minimum distances of the block component codes in the extension field representation Proof: We only need to show that there exists a code- and hence with Theorem 19 a lower bound for the minimum word of C that satisfies this minimum. Up to a permu- distance of the whole code. tation of {1,...,t} we can consider that the code C1 is satisfying the minimum distance. Let g1 ∈ N be such that CONCLUSIONS g1 d(rowsp(U1), rowsp(U1)M e1 ) = d(C1). Since the cardinali- p1 Due to the characterization of conjugacy classes of cyclic ties of the codes Ci are pairwise coprime, it follows that there subgroups of GLn(Fq), we were able to conclude that every exists g ∈ N such that cyclic orbit code is conjugated to a cyclic orbit code defined by the cyclic group generated by a matrix in rational canonical g ≡ g (mod |C |) and g ≡ 0 (mod |C |) 1 1 j form. The research of orbit codes with good parameters can for j =2,...,m. We obtain that then be restricted to this subclass of cyclic orbit codes.

g g1 The following step in this research direction is to completely d(U, UM ) = d(U, Udiag(M e1 ,I,...,I)) p1 classify orbit codes. In order to do so we have to find a g1 = d(rowsp(U1), rowsp(U1)M e1 )= d(C1) characterization of the conjugacy classes of subgroups of p1 GLn(Fq) that possibly coincides with the one presented in Section I if restricted to cyclic subgroups of GLn(Fq). Lemma 21: Let M := diag(M e1 ,...,M et ) ∈ GLn(Fq) p1 pt such that pi ∈ Fq[x] are monic irreducible polynomials and REFERENCES ei ¯ Fki×di di := deg(pi ) for i = 1,...,t. Let ki ≤ di, Ui ∈ q [1] R. Kötter and F. Kschischang, “Coding for errors and erasures in random be matrices with full rank and U := diag(U¯1,..., U¯t) ∈ network coding,” Information Theory, IEEE Transactions on, vol. 54, i N no. 8, pp. 3579–3591, August 2008. GFq (k,n). If we define C := {UM | i ∈ } and Ci := j [2] F. Manganiello, E. Gorla, and J. Rosenthal, “Spread codes and spread {rowsp(U¯iM ei ) | j ∈ N} and it holds gcd(|Ci|, |Cj |)=1 pi decoding in network coding,” in Proceedings of the 2008 IEEE Inter- for all i 6= j, then national Symposium on Information Theory, Toronto, Canada, 2008, pp. 851–855. t [3] A. Kohnert and S. Kurz, “Construction of large constant dimension codes ¯ ¯ j with a prescribed minimum distance,” in MMICS, ser. Lecture Notes in d(C)=2k − 2 max dim rowsp Ui ∩ rowsp UiMpei . X j∈N i Computer Science, J. Calmet, W. Geiselmann, and J. Müller-Quade, Eds., i=1 vol. 5393. Springer, 2008, pp. 31–42. Proof: Also here we show a codeword of C which satisfies [4] T. Etzion and N. Silberstein, “Error-correcting codes in projective spaces via rank-metric codes and ferrers diagrams,” Information Theory, IEEE the relation. Let g1,...,gt ∈ N be such that dim(rowsp(U¯j )∩ ¯ gj Transactions on, vol. 55, no. 7, pp. 2909 –2919, jul. 2009. rowsp(Uj M ej ) is maximal for j = 1,...,m. Since the [5] V. Skachek, “Recursive code construction for random networks,” Infor- pj cardinalities of the codes are pairwise coprime, it follows that mation Theory, IEEE Transactions on, vol. 56, no. 3, pp. 1378 –1382, N March 2010. there exists a g ∈ such that [6] A.-L. Trautmann, F. Manganiello, and J. Rosenthal, “Orbit codes - a new concept in the area of network coding,” in Information Theory Workshop g ≡ gj (mod |Cj |) (ITW), 2010 IEEE, Dublin, Ireland, Aug. 2010, pp. 1 –4. [7] I. N. Herstein, Topics in Algebra, 2nd ed. Lexington, Mass.: Xerox for any j =1,...,t. Then, College Publishing, 1975. [8] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications. Cambridge, London: Cambridge University Press, 1994, g revised edition. dmin(C)= d(U, Udiag(M e1 ,...,M em ) ) p1 pm [9] A.-L. Trautmann and J. Rosenthal, “A complete characterization of g1 gm irreducible cyclic orbit codes,” in Proceedings of the Seventh International = d(U, Udiag(M e1 ,...,Mpem )) p1 m Workshop on Coding and Cryptography (WCC) 2011, 2011, pp. 219 – 223.