Cyclic Orbit Flag Codes
aa r X i v : . [ c s . I T ] F e b Cyclic Orbit Flag Codes
Clementa Alonso-González , Miguel Ángel Navarro-Pérez February 2, 2021
Abstract
In network coding, a flag code is a set of sequences of nested subspacesof F nq , being F q the finite field with q elements. Flag codes defined as orbitsof a cyclic subgroup of the general linear group acting on flags of F nq arecalled cyclic orbit flag codes . Inspired by the ideas in [10], we determine thecardinality of a cyclic orbit flag code and provide bounds for its distancewith the help of the largest subfield over which all the subspaces of a flagare vector spaces (the best friend of the flag). Special attention is paidto two specific families of cyclic orbit flag codes attaining the extremepossible values of the distance: Galois cyclic orbit flag codes and optimumdistance cyclic orbit flag codes . We study in detail both classes of codesand analyze the parameters of the respective subcodes that still have acyclic orbital structure.
Keywords:
Network coding, flag codes, cyclic orbit flag codes.
Network coding is a strong tool for effective data transmission in a networkmodelled as a directed acyclic multigraph with several sources and sinks. In[1], it was proved that the information flow of the network may be improved ifthe intermediate nodes are able to perform random linear combinations of thereceived inputs instead of simply routing them. Random network coding wasintroduced in [12], and an algebraic approach to it was presented in [13]. In thatwork, the authors propose transmitting information by using vector subspaces of F nq and define subspace codes as a class of codes well suited for error correction.In case all the codewords in a subspace code have the same dimension, it issaid to be a constant dimension code . The seminal paper [13] has lately led tomany lines of research on subspace codes addressed either to the constructionof subspace codes with the best size fixed the minimum distance or to findalgebraic constructions of subspace codes with good parameters (see [25] andreferences therein).In [24], Trautmann et al. introduced the concept of orbit codes as subspacecodes obtained from the action of subgroups of the general linear group GL( n, q ) Dpt. de Matemàtiques, Universitat d’Alacant, Sant Vicent del Raspeig, Ap. Correus 99,E – 03080 Alacant.E-mail adresses: [email protected], [email protected] . yclic Orbit Flag Codes on the set of subspaces of F nq . When the acting group is cyclic, we speakabout cyclic orbit codes . This family of codes has awaken a lot of interestdue to the simplicity of their algebraic structure and to the existence of efficientencoding/decoding algorithms. We refer the reader to [5, 6, 8, 9, 10, 19, 21, 23,24, 26] for some of the more recent papers.Taking into account that F nq and the field extension F q n are isomorphic as F q -vector spaces, in [10], the authors consider subspace codes as collections of F q -vector subspaces of F q n and study orbit codes arising from the natural actionof the multiplicative subgroups of F ∗ q n (cyclic groups as well) on F q -vector spaces.Fixed a generating subspace U of the cyclic orbit code Orb( U ) , their main toolis the best friend of U , that is, the largest subfield of F q n over which U is a vectorspace. This concept is closely related with the stabilizer of U , specially whenthe acting group is F ∗ q n . The best friend allows the authors to give relevantinformation about the cardinality, distance and other features of cyclic orbitcodes. Flag codes were introduced in [15] as a generalization of constant dimensioncodes in network coding. In a flag code of constant type, codewords are given bysequences of nested subspaces (flags) of prescribed dimensions. In that paper,the multiplicative action of
GL( n, q ) is naturally extended from subspaces toflags and several constructions of orbit flag codes are provided. In [3, 4], flagcodes attaining the maximum possible distance ( optimum distance flag codes )are characterized and obtained without regard to their possible orbital structurewhereas in [2] an orbital construction of them is proposed.In this work we follow the approach of Gluesing-Luerssen et al. in [10].Inspired by their ideas, we consider flags on F q n given by nested F q -subspacesof the field F q n and focus on cyclic orbit flag codes constructed as orbits ofsubgroups of F ∗ q n . We generalize the concept of the best friend of a subspace tothe flags framework by defining the best friend of a flag as the largest subfieldof F q m over which every subspace in a flag is a vector space. As it occurs inthe constant dimension codes scenario, the knowledge of the best friend of agenerating flag allows us to easily determine the size of the cyclic orbit code aswell as to give estimates for its distance. In particular, we pay special attentionto two specific families of cyclic orbit flag codes attaining the extreme possiblevalues of the distance. We introduce first the concept of Galois cyclic flagcodes as the cyclic orbit codes generated by sequences of nested subfields of F q n . Despite the fact that these codes have the minimum possible distance(fixed the best friend), they present a nice gear of nested spreads compatiblewith the action of F ∗ q n . Moreover, if one consider the subcodes of Galois cyclicflag codes that keep cyclic orbital structure, we can improve their distance in acontrolled manner and reach even the maximum possible one. By the way, wealso determine which dimensions in the type vector of a general generating flagare compatible with attaining the maximum distance, having a fixed best friendand being orbits under the action of subgroups of F ∗ q n . In other words, we study optimum distance cyclic orbit flag codes and their orbital cyclic subcodes.The text is organized as follows. In Section 2, the reader can find the generalbackground on subspace codes. Particular care is devoted to the study of cyclicorbit (subspace) codes developed in [10]. In Section 3, cyclic orbit flag codes2 yclic Orbit Flag Codes are introduced. We also generalize the notions of stabilizer subfield and bestfriend to the flag codes setting by exhibiting the relationship between thesetwo concepts and the corresponding ones for subspace codes. In Section 4, thecardinality and bounds for the distance of a cyclic orbit flag code with a givenbest friend are provided. We finish by introducing Galois cyclic flag codes andoptimum distance cyclic flag codes with a prescribed best friend. We study theirparameters and properties as well as the ones of respective subcodes coming alsofrom the action of subgroups of F ∗ q n . Fix F q the finite field of q elements where q is a primer power. For any naturalnumber n > , F nq represents the n -dimensional vector space over F q . Given k < n , the Grassmannian G q ( k, n ) is the set of k -dimensional subspaces of F nq and we write P q ( n ) to denote the projective geometry of F nq , that is, the setof all the subspaces of F nq . The set P q ( n ) can be considered as a metric spacewith the subspace distance (see [13]) defined as d S ( U , V ) = dim( U + V ) − dim( U ∩ V ) . (1)A subspace code C of length n is a nonempty subset of P q ( n ) and its minimumsubspace distance is defined as d S ( C ) = min { d S ( U , V ) | U , V ∈ C , U 6 = V} . A subspace code in which every codeword has the same dimension, say k ,is called constant dimension code of dimension k and length n (see [25] andreferences therein). The subspace distance between two subspaces U and V ofdimension k is given by d S ( U , V ) = 2( k − dim( U ∩ V )) . Consequently, the minimum distance of a constant dimension code of dimension k is upper bounded by d S ( C ) (cid:26) k if k n, n − k ) if k > n. (2)These bounds for the distance are attained by constant dimension codes in whichevery pair of codewords intersects with the minimum possible dimension. Fordimensions k up to ⌊ n ⌋ , constant dimension codes attaining the previous boundare known as partial spread codes and their cardinality is, at most, ⌊ q n − q k − ⌋ . In contrast, a constant dimension code that attains the bound in (2) and hasdimension k > ⌊ n ⌋ , cannot contain more than ⌊ q n − q n − k − ⌋ elements.A spread code in G q ( k, n ) , or just a k -spread , is a partition of F nq into k -dimensional subspaces. In other words, a spread is a partial spread that cov-ers F nq . Spreads are classical objects coming from Finite Geometry and it iswell known that k -spreads exist if, and only if, k divides n (see [22]). As a3 yclic Orbit Flag Codes consequence, the size of every k -spread is q n − q k − . For further information re-lated to spread codes in the network coding framework, we refer the reader to[11, 16, 17, 25].There are constant dimension codes that can be obtained as orbits of theaction of subgroups of the general linear group
GL( n, q ) on the Grassmannianof the corresponding dimension. In this case, we speak about orbit codes , whichwere introduced for the first time in [24]. Given a k -dimensional subspace U of F nq and a subgroup G of GL( n, q ) , the orbit of U under the action of G isthe constant dimension code given by Orb G ( U ) = {U · A | A ∈ G } , where U · A = rowsp( U A ) for any full-rank generator matrix U of U . The stabilizer of U under the action of G is the subgroup Stab G ( U ) = { A ∈ G | U · A = U } . Clearly, | Orb G ( U ) | = | G || Stab G ( U ) | and its minimum distance is given by d S (Orb G ( U )) = min { d s ( U , U · A ) | A ∈ G \ Stab G ( U ) } . If the group G is cyclic, the code Orb G ( U ) is called cyclic orbit code. Thisspecial family of orbit codes was widely studied in [10, 18, 20, 23]. In particular,using the fact that F nq and F q n are isomorphic as F q -vector spaces, Trautmann et al. provide in [23] the following construction of a k -spread as a cyclic orbitcode. Take a divisor k of n and let α denote a primitive element of F q n , i.e, agenerator of the multiplicative group F ∗ q n . If we put c = q n − q k − , then it is clearthat h α c i is the unique subgroup of order q k − of F ∗ q n and that h α c i∪{ } = F q k . As proved in [23, Th. 31], the stabilizer of F q k under the action of the cyclicgroup h α i is precisely the subgroup h α c i and the orbit S = Orb h α i ( F q k ) = { F q k α i | i = 0 , . . . , c − } (3)is a k -spread of F q n .In [10], Gluesing-Luerssen et al. generalize the construction in (3) for any β ∈ F ∗ q n by introducing the concept of β - cyclic orbit code generated by a subspace U of F q n and study these codes by specifying the largest subfield over which thesubspace U is a vector space. Let us recall some definitions and results fromthat work that we will use along this paper.Consider any nonzero element β in the finite field F q n and the natural mul-tiplicative action of the group h β i on F q -vector subspaces of F q n . Orbits of thisaction are called β - cyclic orbit codes . To be precise, if k < n and U ⊂ F q n is a k -dimensional subspace over F q , the β - cyclic orbit code generated by U isthe constant dimension code in the Grassmannian G q ( k, n ) given by Orb β ( U ) = {U β i | i | β | − } , where | β | denotes the multiplicative order of β (for further information on theseorbits, see [7]). The stabilizer of the subspace U under the action of h β i isthe cyclic subgroup defined as Stab β ( U ) = { β i ∈ h β i | U β i = U } and the stabilizer subfield
Stab + β ( U ) of U (with respect to β ) is the smallest subfield of F q n containing both F q and Stab β ( U ) . yclic Orbit Flag Codes Remark 2.1.
When the acting group is F ∗ q n , following the notation in [10], wesimply write by Orb( U ) and call it the cyclic orbit code generated by U . In thissituation, we also remove the subscript β and write Stab( U ) and Stab + ( U ) tothe denote the stabilizer and the stabilizer subfield of U respectively.Concerning the cardinality of a β -cyclic orbit code, there exists a nice re-lationship between | Orb β ( U ) | and the dimension of the generating subspace U . More precisely, in [10, Prop. 3.7], the authors showed that, if U is a k -dimensional subspace of F q n , then | β q k − | = | β | gcd( | β | , q k − divides | Orb β ( U ) | . (4)Moreover, the equality | Orb β ( U ) | = | β | q k − holds if, and only if, U is a vectorspace over F q k . More precisely, if ∈ U , for every divisor k of n , the code Orb( U ) is a k -spread if, and only if, U = F q k . Therefore, the spread defined in(3) arises as the cyclic orbit code Orb( F q k ) in this context.A subfield F q m of F q n is said to be a friend of a subspace U ⊂ F q n if U is an F q m -vector space. In that case, if t is the dimension of U as F q m -vector space,we have that dim F q ( U ) = mt . Moreover, if { u , . . . , u t } ⊆ U is a basis of U over F q m , then it holds U = F q m u ⊕ · · · ⊕ F q m u t . Note that every subspace U is a vector space over Stab + β ( U ) . In other words, thestabilizer subfield is a friend of U . The largest friend of U is called its best friend (see [10]). The concepts of stabilizer subfield and best friend of a subspace turnto be same in the following situation in which, in addition, the knowledge ofthe best friend of U provides straightforwardly the cardinality of the cyclic orbitcode as well as a lower bound for its distance. Proposition 2.2. ([10, Prop. 3.3, 3.12, 3.13 and 4.1]) If U is a subspace of F q n , then its stabilizer subfield satisfies Stab + ( U ) = Stab( U ) ∪ { } and it contains every friend of U . As a consequence, the field Stab + ( U ) is thebest friend of the subspace U . In particular, if Stab + ( U ) = F q m , then | Orb( U ) | = q n − q m − . Moreover, the value m divides the distance between every pair of subspaces in Orb( U ) and, hence, we have that d S (Orb( U )) > m. Besides, if ∈ U , we havethe inclusion Stab + ( U ) ⊆ U . In classical linear algebra, a flag variety on the field extension F q n is a homoge-neous space that generalizes the Grassmann variety and whose points are flags.5 yclic Orbit Flag Codes The use of flags in network coding was proposed for the first time in [15]. Westart this section by recalling some basic background on flag codes. Next, wewill focus on the family of flag codes that are orbits under the action of a cyclicgroup on the flag variety. Finally, we introduce the concepts of stabilizer sub-field and best friend of a flag, following the ideas in [10], in order to deepen thestructure and properties of the family of cyclic orbit flag codes.
Definition 3.1. A flag F = ( F , . . . , F r ) on F q n is a sequence of nested F q -vector subspaces of F q n , i.e., such that { } ( F ( · · · ( F r ( F nq . The subspace F i is said to be the i -th subspace of F . The type of F is the vector (dim( F ) , . . . , dim( F r )) . In case the type vector is (1 , , . . . , n − , we say that F is a full flag .The flag variety F q (( t , . . . , t r ) , n ) is the set of flags of type ( t , . . . , t r ) on F q n . This variety can naturally be equipped with a metric by extending thesubspace distance defined in (1). Given two flags F = ( F , . . . , F r ) and F ′ =( F ′ , . . . , F ′ r ) in F q (( t , . . . , t r ) , n ) , their flag distance is d f ( F , F ′ ) = r X i =1 d S ( F i , F ′ i ) . Definition 3.2. A flag code of type ( t , . . . , t r ) on F q n is a nonempty subset C ⊆ F q (( t , . . . , t r ) , n ) . Its minimum distance is given by d f ( C ) = min { d f ( F , F ′ ) | F , F ′ ∈ C , F 6 = F ′ } and, in case |C| = 1 , we put d f ( C ) = 0 . For each dimension t i in the type vector of a flag code C , we can associate toit the constant dimension code in the Grassmannian G q ( t i , n ) consisting of theset of the i -th subspaces of flags in C . This set is called the i -projected code of C and we denote it by C i . It is clear that |C i | |C| for every i = 1 , . . . , r . In case |C | = · · · = |C r | = |C| , we say that C is disjoint . As shown in [3], the property ofbeing disjoint is necessary in order to have flag codes that achieve the maximumpossible flag distance. For type ( t , . . . , t r ) , that maximum distance is X t i ⌊ n ⌋ t i + X t i > ⌊ t ⌋ ( n − t i ) (5)and flag codes attaining it are called optimum distance flag codes . In [3, 4]the reader can find constructions of this class of codes as well as the followingcharacterization of them. 6 yclic Orbit Flag Codes Theorem 3.3. [3, Th. 3.11] A flag code is an optimum distance flag code if,and only if, it is disjoint and every projected code attains the maximum possibledistance for its dimension.
As in the case of subspace codes, one can build families of flag codes throughthe action of a group. This approach already appears in [15], where the authorsgeneralize the action of
GL( n, q ) on subspaces of F nq to flags and provide severalconstructions of flag codes as orbits of the action of specific upper unitriangularmatrix groups on the full flag variety.In the next section, following the ideas developed in [10] for subspace codes,we introduce the concept of cyclic orbit flag code as the orbit of the multiplicativeaction of subgroups of F ∗ q n on flags on F q n . Given a nonzero element β in the field F q n , we can extend the natural actionof the cyclic group h β i on F q -subspaces of F q n to flags on F q n as follows. If F = ( F , . . . , F r ) is a flag of type ( t , . . . , t r ) on F q n , we define the flag F β as F β = ( F β, . . . , F r β ) . The set
Orb β ( F ) = {F β j | j | β | − } . (6)is called the β - cyclic orbit flag code generated by F . The stabilizer of the flag F (w.r.t. β ) is the subgroup of h β i given by Stab β ( F ) = { β j ∈ h β i | F β j = F } . When the acting group is F ∗ q n , we do not specify it and simply write Orb( F ) todenote the cyclic orbit flag code generated by F . We also drop the subscript in
Stab( F ) . Observe that every Orb β ( F ) is a subcode of Orb( F ) . Furthermore, itholds Stab β ( F ) = h β i ∩ Stab( F ) . As in the subspace codes framework, the orbital structure simplifies thecomputation of the code parameters: the cardinality of the flag code in (6) isgiven by | Orb β ( F ) | = | β || Stab β ( F ) | = | β ||h β i ∩ Stab( F ) | (7)and its minimum distance can be computed as d f (Orb β ( F )) = min { d f ( F , F β j ) | β j / ∈ Stab β ( F ) } . Remark 3.4.
Notice that the projected codes associated to
Orb β ( F ) are β -cyclic orbit (subspace) codes as well. More precisely, for every i r , wehave (Orb β ( F )) i = Orb β ( F i ) . yclic Orbit Flag Codes Moreover, as for any other group action, it holds a clear relationship betweenthe stabilizer of the flag F and the ones of its subspaces: Stab β ( F ) = r \ i =1 Stab β ( F i ) . (8)This equality leads to a direct link between the cardinality of a β -cyclic orbitflag code, the ones of its projected codes, and the dimensions on the generatingflag type vector. Proposition 3.5.
Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t r ) on F q n and β ∈ F ∗ q n . Then | Orb β ( F i ) | divides | Orb β ( F ) | , for i r . In particular, lcm n | β q ti − | | i r o divides | Orb β ( F ) | . Proof.
Recall that | Orb β ( F ) | = | β || Stab β ( F ) | and | Orb β ( F i ) | = | β || Stab β ( F i ) | , for eve-ry i r . Moreover, by means of (8), we have that | Stab β ( F ) | divides | Stab β ( F i ) | for every value of i . Hence, the cardinality of Orb β ( F i ) must divide | Orb β ( F ) | , for i r . The last part of the statement follows directly fromthis fact along with (4). The following definition extends the concept of stabilizer subfield of a subspacedefined in [10] to the flag codes setting.
Definition 3.6.
Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t r ) on the field F q n and β ∈ F ∗ q n . We define the stabilizer subfield of the flag F (w.r.t. β ) asthe smallest subfield Stab + β ( F ) of F q n containing both F q and Stab β ( F ) .As before, if β is a primitive element of F q n , we just write Stab + ( F ) . In thiscase, the stabilizer subfield of a flag admits the following nice description: Proposition 3.7.
Let F = ( F , . . . , F r ) be a flag on F q n . It holds Stab + ( F ) = Stab( F ) ∪ { } = r \ i =1 Stab + ( F i ) and every i -th subspace F i of the flag F is a vector space over Stab + ( F ) . More-over, if ∈ F , the stabilizer subfield Stab + ( F ) is contained in every subspaceof F .Proof. By application of Proposition 2.2, one has that
Stab + ( F i ) = Stab( F i ) ∪{ } for every i r . Now, by means of (8), we conclude that Stab( F ) ∪ { } = ( T ri =1 Stab( F i )) ∪ { } = ( T ri =1 Stab( F i ) ∪ { } ) = T ri =1 Stab + ( F i ) . (9)8 yclic Orbit Flag Codes This proves that
Stab( F ) ∪ { } is a field and then it is the stabilizer subfield ofthe flag F . Moreover, it is a subfield of every Stab + ( F i ) . Hence, it is clear thatthe subspace F i is a vector space over Stab + ( F ) . Besides, if ∈ F , by usingProposition 2.2, we obtain Stab + ( F ) ⊆ Stab + ( F ) ⊆ F ⊂ F ⊂ · · · ⊂ F r . Notice that the condition ∈ F in Proposition 3.7 is by no means restrictivewhen the acting group is F ∗ q n . In fact, we can always find a generating flagfulfilling this property. It suffices to see that, given an arbitrary flag F , forevery nonzero element β ∈ F , the flag F β − clearly satisfies the requiredcondition. Moreover, since β is an element in the field F ∗ q n , both flags F and F β − generate the same cyclic orbit flag code Orb( F ) = Orb( F β − ) . Remark 3.8.
Clearly, if β ∈ F ∗ q n , it holds Stab β ( F ) ⊆ Stab( F ) and, hence, Stab + β ( F ) ⊆ Stab + ( F ) . As a consequence, every F i is a vector space overthe field Stab + β ( F ) as well as over all its subfields. Moreover, if ∈ F , then Stab + β ( F ) ⊆ F i for i r .As it occurs for constant dimension codes, the inclusion Stab + β ( F ) ⊆ Stab + ( F ) may be strict. Let us provide an example from a length-two flag inspired by[10, Example 3.6]. Example 3.9.
Consider the flag F = ( F , F ) on the field F and let α be aprimitive element of F . Observe that Stab + ( F ) = F and Stab + ( F ) = F .Hence, by Proposition 3.7, it follows that Stab + ( F ) = F ∩ F = F . Let usnow choose β = α , which have multiplicative order equal to . Observe that Stab β ( F ) ⊆ h β i and also Stab β ( F ) ⊆ Stab + β ( F ) ∗ ⊆ Stab + ( F ) ∗ = F ∗ . As theorders of h β i and F ∗ are coprime, we have that Stab β ( F ) = { } . This impliesthat Stab + β ( F ) = F .There are remarkable connections between the cardinality of a β -cyclic or-bit flag code and the generating flag when one has a divisor of n among thedimensions of the type vector. Proposition 3.10.
Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t t ) on F q n .Assume that m is a divisor of n such that m = t i for some i ∈ { , . . . , r } andconsider the subfield F q m of F q n . Take an element β ∈ F ∗ q n such that F ∗ q m ⊆ h β i .Then:(1) The value | β | q m − divides | Orb β ( F ) | . (2) We have | Orb β ( F ) | = | β | q m − if, and only if, each subspace F j is a vectorspace over F q m . In particular, t = m .Proof. As F ∗ q m ⊆ h β i , we have that q m − must divide | β | . This implies that | β q ti − | = | β q m − | = | β | q m − and (1) follows directly from Proposition 3.5.9 yclic Orbit Flag Codes To prove (2), observe that | Orb β ( F ) | = | β | q m − holds if, and only if, Stab β ( F ) is a subgroup of order q m − of h β i . By the uniqueness of subgroups of a cyclicgroup, it follows that Stab β ( F ) = F ∗ q m . Hence, the field Stab + β ( F ) = F q m isa subfield of Stab + ( F ) and, by means of Remark 3.8, every subspace F j hasstructure of F q m -vector space. In particular, no dimension smaller than m canappear in the type vector, i.e., t = m .Conversely, assume that every F j is a vector space over F q m for j ∈ { , . . . , r } .In particular, F = F q m γ for some γ ∈ F ∗ q n . As a consequence, multiplicationby elements in F ∗ q m ⊆ h β i is closed on every subspace F j . Hence, we have F ∗ q m ⊆ Stab β ( F j ) for j r and, by means of (8), it holds F ∗ q m ⊆ Stab β ( F ) .On the other hand, notice that Stab β ( F ) ⊆ Stab β ( F ) = F ∗ q m . Thus, it followsthat Stab β ( F ) = F ∗ q m and | Orb β ( F ) | = | β | q m − , as we wanted to prove.The second statement in Proposition 3.10 turns out specially interesting inthe case of cyclic orbit codes, that is, when the acting group is F ∗ q n . Corollary 3.11.
Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t r ) on F q n .Assume that m is a divisor of n such that m = t i for some i ∈ { , . . . , r } . If | Orb( F ) | = q n − q m − , then m = t and the constant dimension code Orb( F ) is the m -spread Orb( F q m ) . Moreover, the value m divides t j , for j ∈ { , . . . , r } .Proof. By means of Proposition 3.10, it is clear that the first dimension in thetype vector is t = m and it divides every t i . Moreover, F must be a one-dimensional vector space over F q m , that is, it is of the form F = F q m γ forsome γ ∈ F ∗ q n . As a result, the first projected code Orb( F ) = Orb( F q m ) is the m -spread defined in (3). Remark 3.12.
In the conditions of the previous corollary, if we require thesubspace F to contain the element ∈ F q n , not only do we obtain that Orb( F ) = Orb( F q m ) but also the equality F = F q m .In view of Propositions 3.7 and 3.10, it also makes sense the extension toflags of the concept of best friend introduced in [10]. Definition 3.13.
Consider a flag F on F q n . A subfield F q m of F q n is said to bea friend of the flag F if all its subspaces are F q m -vector spaces. In other words,a subfield of F q n is a friend of the flag F if it is a friend of all its subspaces. Wecall best friend of the flag F to its largest friend.The next result states a necessary condition on the type vector of flags havinga given subfield of F q n as a friend. The proof is straightforward. Lemma 3.14.
Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t r ) on F q n . If F q m is a friend of F then m divides gcd( t , . . . , t r , n ) . Remark 3.15.
If follows that the best friend of a flag of type ( t , . . . , t r ) with gcd( t , . . . , t r , n ) = 1 , in particular a full flag, is the ground field F q .Beyond conditions on the type vector, we can always characterize the bestfriend of an arbitrary flag in terms of the ones of its subspaces. To do so, wegeneralize Proposition 2.2 to the flag codes scenario.10 yclic Orbit Flag Codes Proposition 3.16.
Let F = ( F , . . . , F r ) be a flag on F q n . Then Stab + ( F ) isthe best friend of the flag F and it contains any other friend F q m of F . Moreover,if ∈ F , then we have that F q m ⊆ Stab + ( F ) ⊆ F .Proof. Let us prove that
Stab + ( F ) is the largest friend of F , i.e., its best friend.To do so, assume that a subfield F q m of F q n is a friend of the flag F . By definitionof friend of a flag, we know that multiplication by elements in F q m is closed inevery subspace F i of the flag. As a consequence, F ∗ q m is a subgroup of Stab( F ) and we can conclude that F q m is contained in Stab( F ) ∪ { } = Stab + ( F ) . Thisproves that the stabilizer subfield of F is its best friend. Finally, by using thecondition ∈ F together with Proposition 3.7, we obtain the inclusion F q m ⊆ Stab + ( F ) ⊆ Stab + ( F ) ⊆ F . Remark 3.17.
Observe that all flags in the code
Orb( F ) have the same bestfriend. In particular, since Orb β ( F ) ⊆ Orb( F ) , flags in a β -cyclic orbit flag codehave all the same best friend for every β ∈ F ∗ q n . Hence, we say that Stab + ( F ) is the best friend of every Orb β ( F ) .As stated in the proof of Proposition 3.7, (see equation (9)), the stabilizersubfield of the cyclic flag code Orb( F ) can be computed as the intersection ofthe ones of its projected codes. Combining this with Proposition 3.16, we obtainthe next result. Corollary 3.18.
Let F = ( F , . . . , F r ) be a flag on F q n . Then its best friend isthe intersection of the ones of its subspaces. Moreover, if ∈ F , every friendof the flag F is contained in F . It is clear that the best friend of a flag is a subfield of the ones of its subspaces.However, while the subspaces in a flag are nested, their respective best friendsmight not form a sequence of nested subfields as we can see in the followingexample.
Example 3.19.
Take q a prime power and the flag of type (2 , on F q givenby F = ( F q , F q + F q α ) , where α denotes a primitive element of F q . In thiscase, the best friend of F is precisely F q whereas, since gcd(3 ,
4) = 1 , the bestfriend of F is the ground field F q .As it happens in the subspace codes setting, knowing the best friend of acyclic orbit flag code gives relevant information about the code parameters aswe will see below. This section is devoted to the study of cyclic orbit flag codes on F q n generatedby flags with the subfied F q m as their best friend. From now on, the integer m will denote a divisor of n . Let us first see how the close relationship betweenthe best friend of a flag and its stabilizer allows us to compute the size of thegenerated cyclic or β -cyclic orbit flag code. The next result follows from (7) andProposition 3.16. 11 yclic Orbit Flag Codes Proposition 4.1.
Let F = ( F , . . . , F r ) be a flag on F q n and β ∈ F ∗ q n . Assumethat F q m is the best friend of F . Then | Orb β ( F ) | = | β ||h β i ∩ F ∗ q m | . In particular, if β is a primitive element of F q n , it holds | Orb( F ) | = q n − q m − . Remark 4.2.
It is well known that any orbit coming from the action of agroup can be partitioned into a set of orbits when we restrict the action to asubgroup. These orbits may have different cardinality in general. However, thecardinality of the code
Orb β ( F ) just depends on | β | and the best friend of F .Moreover, since all the flags in Orb( F ) have the same best friend, we have that | Orb β ( F ′ ) | = | Orb β ( F ) | for every F ′ ∈ Orb( F ) . We conclude that, for any β ∈ F ∗ q n the code Orb( F ) can be partitioned into a set of β -cyclic subcodes, allof them with the same cardinality.Proposition 4.1 leads to a characterization of β -cyclic orbit flag codes whosesize coincides with the order or the acting group. Corollary 4.3.
Let F be a flag on F q n with F q m as its best friend and consider β ∈ F ∗ q n . Then | Orb β ( F ) | = | β | if, and only if | β | and q m − are coprime. Inparticular, this equality always holds if q = 2 and m = 1 . Having the subfield F q m as best friend yields a condition on the type vector ofa flag, as well as a description of the structure of all the flags in its β -cyclic orbitflag code in terms of F q m . Let F = ( F , . . . , F r ) be a flag of type ( t , . . . , t r ) on F q n with F q m as its best friend. Hence, F q m must be a friend of all its subspacesand m divides every dimension in the type vector. Consequently, we can write t i = ms i for i = 1 , . . . , r , where s < s < · · · < s r < s = nm . On the otherhand, the nested structure of the flag F allows us to find linearly independentelements a , . . . , a s r ∈ F q n (over F q m ) such that, for every i r , we have F i = s i M j =1 F q m a j . In particular, observe that if m is a dimension in the type vector, then s = 1 andthe cyclic orbit code Orb( F ) is the m -spread of F q n described in (3). Moreover,if ∈ F , this subspace must be the subfield F q m .Concerning the distance of β -cyclic orbit flag codes, as in the constant dimen-sion codes framework, we can also deduce some estimates from the knowledgeof the best friend. Proposition 4.4.
Let F be a flag of type ( ms , . . . , ms r ) on F q n with the subfield F q m as its best friend and take β ∈ F ∗ q n . Then d f (Orb β ( F )) = 0 if, and only if, β ∈ F ∗ q m . Out of this case, m divides d f (Orb β ( F )) and it holds m d f (Orb β ( F )) m X s i ⌊ s ⌋ s i + X s i > ⌊ s ⌋ ( s − s i ) . (10)12 yclic Orbit Flag Codes Proof.
Assume that d f (Orb β ( F )) = 0 or, equivalently, that Orb β ( F ) = {F } .This happens if, and only if, β stabilizes the flag F , i.e., if β ∈ Stab( F ) = F ∗ q m .Take now β ∈ F ∗ q n \ F ∗ q m . By the definition of best friend of the flag F , itfollows that F q m is a friend of every subspace F i . This implies that, for every i r , subspaces in Orb β ( F i ) are vector spaces over F q m . Take a flag F ′ in Orb β ( F ) \ {F } . Since, for every i r, both F i , F ′ i , and hence F i ∩ F ′ i , arevector spaces over F q m , the value m divides their dimensions (over F q ). Takinginto account that d S ( F i , F ′ i ) = 2(dim( F i ) − dim( F i ∩ F ′ i )) , we conclude that m divides d S ( F i , F ′ i ) for every i r . Consequently, the value m alsodivides d f ( F , F ′ ) = P ri =1 d S ( F , F ′ ) , for every choice of F ′ ∈ Orb β ( F ) \ {F } .In particular, m divides d f (Orb β ( F )) and it is a lower bound for it. At thesame time, if we consider the general upper bound for the distance of flag codesof type ( ms , . . . , ms r ) on F q n given in (5), taking into account that n = ms ,we obtain the result. Remark 4.5.
Notice that for every β ∈ F ∗ q n , it holds Orb β ( F ) ⊆ Orb( F ) . Thenit follows d f (Orb β ( F )) > d f (Orb( F )) except for β ∈ Stab( F ) = F ∗ q m . However,not every β allows us to improve the distance with respect to the one of Orb( F ) .We can appreciate this fact in the next example. Example 4.6.
Take q a prime power and α a primitive element of F q . Consider F a flag of type (1 , on F q with subspaces F = F q and F = F q + F q α. Notice that, since gcd(1 , ,
6) = 1 , by application of Lemma 3.14, F q is the bestfriend of F . Clearly, it is the best friend of F as well. Concerning F , observethat F q is one of its friends. Hence, its best friend is a subfield of F q containing F q . We conclude that F q is the best friend of F . The cyclic orbit flag code Orb( F ) contains exactly q − q − flags and we have d f (Orb( F )) = 2 . It suffices tosee that, for every β ∈ F ∗ q \ F ∗ q ⊂ F ∗ q , it holds F = F β and d f ( F , F β ) = d S ( F , F β ) = 2 . Observe that this is the minimum possible distance fixed the best friend F q .Now, if we consider the subgroup h γ i = F ∗ q , the subcode Orb γ ( F ) has cardi-nality q − q − = q + 1 and the same argument above gives that d f (Orb γ ( F )) = 2 .In this case, Orb γ ( F ) does not have a better distance than Orb( F ) . Take now δ ∈ F ∗ q a generator of F ∗ q , then the δ -cyclic flag code generated by F contains q − q − = q + q + 1 flags. To compute its distance, observe that Stab δ ( F ) = h δ i ∩ F ∗ q = F ∗ q ∩ F ∗ q = F ∗ q = Stab δ ( F ) = Stab δ ( F ) . Hence, for every δ i / ∈ Stab δ ( F ) it holds F j = F j δ i , for j = 1 , . On the one hand,we have d S ( F q , F q δ i ) = 2 . On the other hand, as F q is the best friend of F ,the value d S ( F , F δ i ) is a multiple of . Since the maximum possible distancebetween -dimensional subspaces of F q is precisely −
4) = 4 , it follows that d S ( F , F δ i ) = 4 . As a result, d f ( F , F δ i ) = 6 for all δ i ∈ h δ i \ Stab δ ( F ) and weconclude that d f (Orb δ ( F )) = 6 > d f (Orb( F )) . yclic Orbit Flag Codes Remark 4.7.
Observe that the upper bound for the distance given in (10) co-incides with the general bound for the flag distance given in (5). However, inSubsection 4.2, we will see that, in our scenario, not every type vector is com-patible with attaining this upper bound. On the other hand, the lower boundfor the distance of a β -cyclic flag code having F q m as its best friend obtained in(10) coincides with the one given in Proposition 2.2 for cyclic (subspace) codeshaving the same best friend. The previous example shows that this lower boundcan also be attained by β -cyclic obit flag codes of length at least two. Let ussee another situation where the generating flag has a special form. Example 4.8.
Let F = ( F , F ) be the flag of type (2 , on F definedin Example 3.9 and consider the cyclic orbit flag code Orb( F ) . Observe that,as stated in 3.9, the best friend of the flag F is the subfield F . Moreover, Stab( F ) = Stab( F ) = F and Stab( F ) = F . Now, if α denotes a primitiveelement of F , the power α is also a primitive element of the subfield F .Hence, α clearly lies in F ∗ \ F ∗ . As a result, the flags F and F α are differentcodewords in Orb( F ) whereas we have the subspaces equality F = F α . Itfollows that d f (Orb( F )) d f ( F , F α ) = d S ( F , F α ) = 4 , which is the minimum possible distance between subspaces of dimension oneover F . Hence, we conclude that d f (Orb( F )) = 4 .Notice that in the previous example the two subspaces of the generatingflag are nested subfields of a given finite field. This example gives rise to thedefinition of a family of cyclic orbit flag codes inspired by the towers of subfieldsof F q n . Let t < · · · < t r < n be a sequence of divisors of n such that t i divides t i +1 , for i r − . Definition 4.9.
We define the
Galois flag of type ( t , . . . , t r ) on F q n as the flaggiven by the sequence of nested subfields ( F q t , . . . , F q tr ) . For every β ∈ F ∗ q n ,the β -cyclic orbit flag code generated by this flag is called the Galois β -cyclicflag code of type ( t , . . . , t r ) . When β is primitive, we just say Galois cyclic flagcode.
Remark 4.10.
Notice that, for each subgroup h β i ⊆ F ∗ q n , there is just oneGalois β -cyclic flag code for each type vector satisfying the condition above.In contrast, the Galois β -cyclic flag code of a fixed type can be generated bydifferent flags consisting of sequences of subspaces, not necessarily fields. Never-theless, if we impose the condition ∈ F , only the Galois flag of type ( t , . . . , t r ) can generate the Galois β -cyclic flag code of this type.Given the Galois flag F of type vector ( t , . . . , t r ) , it is clear that its i -thsubspace is its own best friend. Hence, contrary to what happens for generalflags (see Example 3.19), the best friends of the Galois flag subspaces form a14 yclic Orbit Flag Codes sequence of nested subfieds. As a consequence, the first subfield F q t is the bestfriend of the Galois flag of type ( t , . . . , t r ) and, in order to construct Galois β -cyclic flag codes with the subfield F q m as its best friend, it suffices to considera sequence of suitable divisors ( t , . . . , t r ) starting at t = m .Let us start focusing on Galois cyclic flag codes ( β primitive). According toProposition 4.1, the cardinality of the Galois cyclic flag code of type ( t , . . . , t r ) is c = ( q n − / ( q t − whereas its distance is t . In particular, its i -projectedcode contains exactly c i = ( q n − / ( q t i − subspaces and has subspace distanceequal to t i . In spite of the fact that the distance of Galois cyclic flag codesis the smallest possible for cyclic orbit flag codes with a fixed best friend, thekaleidoscopic algebraic structure of nested spreads inside them is remarkableand deserves to be pointed out. Theorem 4.11.
Let F = ( F q t , . . . , F q tr ) be the Galois flag of type ( t , . . . , t r ) on the field F q n and Orb( F ) the associated Galois cyclic flag code. Consider α and α i respective primitive elements of the fields F q n and F q ti , for i r .Then it holds:(1) Each projected code of Orb( F ) is a t i -spread of F q n .(2) The α j -cyclic orbit code Orb α j ( F q ti α l ) is a t i -spread of the subspace F q tj α l ,for every i < j r and l c j − , where c j = ( q n − / ( q t j − .Proof. Observe that, by the definition of Galois cyclic flag code, the i -projectedcode Orb( F i ) = Orb( F q ti ) is the t i -spread of the field F q n described in (3). Thesame argument allows us to state that, for every i < j r , the α j -cyclic orbitcode Orb α j ( F q ti ) is a t i -spread of F q tj as well. Moreover, since the subspacedistance is invariant by the multiplicative action of F ∗ q n = h α i on subspaces, wehave that Orb α j ( F q ti α l ) is also a t i -spread of the vector space F q tj α l , for every l q n − . Now, taking into account that α c j is a primitive element of F q tj , we have that h α j i = h α c j i = F ∗ q tj and F q tj = F q tj α c j . This fact allows usto restrict ourselves to exponents l c j − . Remark 4.12.
Note that Theorem 4.11, describes a striking cyclic spreads gear.First, every projected code of a Galois cyclic flag code is a spread. Then, everycodeword in the j -projected code Orb( F q tj ) , i.e., every subspace of the form F q tj α l , is partitioned into the subspaces of the α j -cyclic orbit code Orb α j ( F q ti α l ) if i < j r . Thereby, we have that Orb α j ( F q ti α l ) is a t i -spread of F q tj α l forevery value l c j − and also a partial spread of dimension t i of the field F q n . Finally, the union of all these orbits ˙ [ c j − l =0 Orb α j ( F q ti α l ) gives us back the t i -spread Orb( F q ti ) = Orb( F i ) . In other words, Galois cyclicflag codes provide collections of nested spreads that respect the orbital structureinduced by the action of h α i on flags. 15 yclic Orbit Flag Codes ...... ... ............ α α α t t t Figure 1: Nested spread structure of a Galois cyclic flag codeThe previous figure represents the structure of the Galois cyclic flag code of agiven type ( t , t , t ) . Vertices are subspaces, (directed) edges denote inclusions(from left to right) and flags are given by directed paths in the graph. Eachcolumn in the graph is a projected code and, by Theorem 4.11, all of them arespreads of F nq of the corresponding dimensions. In addition, every subspace inthe graph is partitioned into the set of its left adjacent vertices. On the otherhand, the Galois flag F = ( F q t , F q t , F q t ) is represented by the sequence of redvertices. Since Stab( F ) = F ∗ q t = h α i , the code Orb α ( F ) consists of the singleelement F . In contrast, for i = 2 , , the code Orb α i ( F ) is given by the set offlags in the graph marked by the round arrow labeled with α i .Take now an element β ∈ F ∗ q n . Let F be the Galois flag of type ( t , . . . , t r ) on F q n and consider the Galois β -cyclic flag code Orb β ( F ) . Since F q t is thebest friend of F , it follows that Stab β ( F ) = h β i ∩ F ∗ q t . Moreover, for everyvalue of i r , it holds Stab β ( F i ) = h β i ∩ F ∗ q ti . As a result, we have thefollowing sequence of nested subgroups of h β i Stab β ( F ) = Stab β ( F ) ⊆ Stab β ( F ) ⊆ · · · ⊆ Stab β ( F r ) ⊆ h β i . (11)By means of Proposition 4.1, the cardinality of Orb β ( F ) and the one of its i -projected code, for every i r , are respectively | Orb β ( F ) | = | β ||h β i ∩ F ∗ q t | and | Orb β ( F i ) | = | β ||h β i ∩ F ∗ q ti | . Furthermore, from Theorem 4.11, and taking into account that
Orb β ( F ) ⊆ Orb( F ) , we can derive the following result for the projected codes of a Galois β -cyclic flag code. Corollary 4.13.
Let F = ( F q t , . . . , F q tr ) be the Galois flag of type ( t , . . . , t r ) on the field F q n and take a nonzero element β ∈ F q n . For each i r, yclic Orbit Flag Codes we write β i to denote a generator of the cyclic subgroup h β i i = Stab β ( F q ti ) = h β i ∩ F ∗ q ti . Then the following statements hold:(1) The projected code
Orb β ( F i ) is a partial spread of dimension t i of F q n .(2) The β j -cyclic orbit code Orb β j ( F q ti β l ) is a partial spread of dimension t i of the subspace F q tj β l , for every i < j r and l | β j | − . Concerning the distance of Galois β -cyclic flag codes, since they are subcodesof the Galois cyclic flag code of the same type, their distance might be betterthan t , apart from the case of the trivial subcode consisting just of the Galoisflag, which has distance equal to zero. Actually, it is possible to determine theexact distance of a Galois β -cyclic flag code by checking the relationship betweenthe subgroup h β i and the subfields F q ti and vice versa, that is, if we choose apermitted distance, we can find a suitable subgroup (possibly not unique) tobuild a β -cyclic orbit Galois attaining such a distance. We state the preciseconditions in the following result: Theorem 4.14.
Let F be the Galois flag of type ( t , . . . , t r ) and consider anelement β ∈ F ∗ q n . Then d f (Orb β ( F )) ∈ { , t , t + t ) , . . . , t + t + · · · + t r ) } .Moreover,(1) d f (Orb β ( F )) = 0 if, and only if, Stab β ( F ) = Stab β ( F r ) = h β i .(2) d f (Orb β ( F )) = 2 P ri =1 t i if, and only if, Stab β ( F ) = Stab β ( F r ) = h β i .(3) d f (Orb β ( F )) = 2 P j − i =1 t i if, and only if, Stab β ( F ) = Stab β ( F r ) and j ∈ { , . . . , r } is the minimum index such that Stab β ( F ) ( Stab β ( F j ) . Proof.
Recall that for every choice of β , the projected codes of Orb β ( F ) are par-tial spreads. As a result, for every l | β | − , we have that d S ( F j , F j β l ) ∈{ , t j } . Moreover, d S ( F j , F j β l ) = 0 holds if, and only if, β l ∈ Stab β ( F j ) . Inthis case, since Stab β ( F j ) ⊆ · · · ⊆ Stab β ( F r ) by (11), we have d S ( F i , F i β l ) = 0 , for every j i r . Hence, distances between flags in Orb β ( F ) belong to theset { , t , t + t ) , . . . , t + t + · · · + t r ) } . Let us see that all of them canbe reached, by showing (1), (2) and (3).(1) As proved in Proposition 4.4, we have d f (Orb β ( F )) = 0 if, and only if, β ∈ F ∗ q t = Stab( F ) or, by using (8), β ∈ Stab( F i ) for all i r . Since Stab β ( F i ) = h β i ∩ Stab( F i ) is always a subgroup of h β i , the previouscondition is equivalent to Stab β ( F i ) = h β i , for every i r . Hence, by(11), we just need to check the equality Stab β ( F ) = Stab β ( F r ) = h β i . In the remaining cases,
Stab β ( F ) must be a proper subgroup of h β i .(2) Assume now that d f (Orb β ( F )) = 2 P ri =1 t i . Hence, for every β l ∈ h β i \ Stab β ( F ) , it must hold d S ( F i , F i β l ) = 2 t i , for all i r . Thishappens if, and only if, β l / ∈ Stab β ( F i ) for every i r . As aconsequence, Stab β ( F i ) ⊆ Stab β ( F ) . On the other hand, by (8), weconclude that Stab β ( F ) = Stab β ( F i ) for every i r . Again,since these stabilizer subgroups are nested, this condition is equivalentto Stab β ( F ) = Stab β ( F r ) . yclic Orbit Flag Codes (3) Consider the case d f (Orb β ( F )) = 2 P j − i =1 t i for some j r . In otherwords, there exists some β l ∈ h β i \ Stab β ( F ) such that d f (Orb β ( F )) = d f ( F , F β l ) = 2 j − X i =1 t i . This happens if, and only if d S ( F i , F i β l ) = (cid:26) t i if i j − , if j i r, or equivalently, if β l ∈ h β i\ Stab β ( F i ) for i j − , and β l ∈ Stab β ( F i ) for j i r . Hence, we conclude Stab β ( F ) = Stab β ( F ) = · · · = Stab β ( F j − ) ( Stab β ( F j ) . Graphically, Galois β -cyclic flag codes can be represented as subgraphs ofthe graph in Figure 1. In the next picture, flags in a Galois β -cyclic flag codeare marked with black lines. In contrast, directed paths containing dotted edgesrepresent flags in Orb( F ) \ Orb β ( F ) . The index j in Theorem 4.14 states thatno flags in the code share subspaces of dimensions t i , for i j − , whereasthere exist different flags having the same j -th subspace. At left, and example ofGalois β -cyclic flag code with distance t ( j = 2 ). At right, the correspondingindex and distance are j = 3 and t + t ) , respectively....... ... ............ t t t ...... ... ............ t t t Figure 2: Two different Galois β -cyclic of type ( t , t , t ) . Observe that Theorem 4.14 allows us to provide specific constructions ofGalois β -cyclic flag codes with a prescribed distance just by choosing a suitableelement β ∈ F ∗ q n . Moreover, since F ∗ q n = h α i , being α a primitive element of F q n , we can translate the above conditions on the stabilizers (w.r.t. β ) in termsof suitable powers of α as follows. Given β ∈ F ∗ q n , we can write | β | = ( q n − /l for some divisor l of q n − . Hence, by the uniqueness of subgroups of a givenorder of the cyclic group F ∗ q n , it is clear that h β i = h α l i . In particular, if18 yclic Orbit Flag Codes c i = ( q n − / ( q t i − , we have that F ∗ q ti = h α c i i , for every i r . Asa consequence, it holds Stab β ( F q ti ) = h β i ∩ F ∗ q ti = h α l i ∩ h α c i i = h α l i i , where l i = lcm( l, c i ) . Moreover, given that each c i +1 divides c i , then l i +1 divides l i , forevery i r − , and the sequence of nested stabilizers given in (11) becomes h α l i ⊆ h α l i ⊆ · · · ⊆ h α l r i ⊆ h α l i . Now, since l, c , . . . , c r divide q n − , every exponent l i divides q n − as well.Hence, the order of each stabilizer is | Stab β ( F q ti ) | = | α l i | = q n − l i , for every i r . We can reformulate Theorem 4.14 as follows: Theorem 4.15.
Let F be the Galois flag of type ( t , . . . , t r ) and consider β ∈ F ∗ q n such that h β i = h α l i for some divisor l of q n − . It holds:(1) d f (Orb β ( F )) = 0 if, and only if, l = l r = l .(2) d f (Orb β ( F )) = 2 P ri =1 t i if, and only if, l = l r = l .(3) d f (Orb β ( F )) = 2 P j − i =1 t i if, and only if, l = l r and j r is theminimum index such that l = l j Example 4.16.
Take F the Galois flag of type (2 , , on F and let α be a primitive element of F . The following table shows the parameters ofall possible Galois β -cyclic flag codes of this type. The sizes of the stabilizersubgroups (w.r.t. β ) of the fields F , F and F are given, together with thecardinality and distance (just denoted by d β ) of Orb β ( F ) . β | β | | Stab β ( F ) | | Stab β ( F ) | | Stab β ( F ) | | Orb β ( F ) | d β α α α α α α α
771 3 3 3 257 28 α
257 1 1 1 257 28 α
255 3 15 255 85 4 α
85 1 5 85 85 4 α
51 3 3 51 17 12 α
17 1 1 17 17 12 α
15 3 15 15 5 4 α α Table 1: Parameters of all Galois β -cyclic flag codes of type (2 , , on F .Clearly, different subgroups of F ∗ q n can provide the same code. For instance,the subgroup h α i gives the Galois cyclic flag code Orb( F ) . We have also Orb α ( F ) = Orb α ( F ) or Orb α ( F ) = Orb α ( F ) , among other possibilities.19 yclic Orbit Flag Codes Remark 4.17.
As proved in the previous theorem, the Galois β -cyclic code oftype ( t , . . . , t r ) attains the maximum possible distance for its type if, and onlyif, it holds Stab β ( F ) = Stab β ( F r ) ( h β i . (12)In other words, if condition (12) is satisfied, we can build an optimum distanceflag code with F q t as its best friend. This fact drives us to investigate cyclicorbit flag codes with the maximum possible distance and fixed best friend whenthe generating flag is not necessarily a Galois flag. This subsection is devoted to the study of flag codes on F q n reaching themaximum distance and being also β -cyclic orbit flag codes with a prescribedbest friend F q m . To tackle this problem, we have to take into account firstthat, in particular, optimum distance flag codes must be disjoint as provedin [3]. Recall that a flag code C of type ( t , . . . , t r ) is said to be disjoint if |C| = |C | = · · · = |C r | . In our specific context, we have that a β -cyclic flag code Orb β ( F ) is disjoint if, and only if, | β || Stab β ( F ) | = | β || Stab β ( F ) | = · · · = | β || Stab β ( F r ) | or, equivalently, if all the stabilizers Stab β ( F ) , Stab β ( F ) , . . . , Stab β ( F r ) havethe same order. In fact, by the uniqueness of subgroups of a cyclic group, allthese stabilizers must coincide. Moreover, by using (8), we have the next result: Proposition 4.18.
The following statements are equivalent:(1)
Orb β ( F ) is a disjoint flag code,(2) Stab β ( F ) = Stab β ( F ) = · · · = Stab β ( F r ) .(3) Stab β ( F ) = · · · = Stab β ( F r ) . In light of Propositions 3.7 and 3.16, the best friend of a flag F can becomputed as Stab + ( F ) = Stab( F ) ∪ { } . Similarly, the best friend of its sub-spaces are given by Stab + ( F i ) = Stab( F i ) ∪ { } . The next result leads directlya characterization of disjoint β -cyclic orbit flag codes in terms of β and the bestfriends of the generating flag and its subspaces. Proposition 4.19.
Let F = ( F , . . . , F r ) be a flag on F q n with F q m as its bestfriend and take β ∈ F ∗ q n . If F q mi denotes the best friend of F i , then the β -cyclicorbit code Orb β ( F ) is disjoint if, and only if h β i ∩ F ∗ q m = h β i ∩ F ∗ q m = · · · = h β i ∩ F ∗ q mr . In particular, the cyclic orbit flag code
Orb( F ) is disjoint if, and only if, all thesubspaces in the flag have the field F q m as their best friend. yclic Orbit Flag Codes Proof.
By means of Proposition 4.18, the code
Orb β ( F ) is disjoint if, and onlyif, for every i r , it holds Stab β ( F i ) = Stab β ( F ) . Since Stab β ( F i ) = h β i ∩ F ∗ q mi , for every i r , and Stab β ( F ) = h β i ∩ F ∗ q m , the result follows.In the particular case of β primitive, then it must hold Stab( F i ) = F ∗ q m , i.e., thebest friend of each F i coincides with the one of F .Observe that it is possible to give a tighter lower bound for the distance ofdisjoint β -cyclic orbit flag codes with F mq as best friend. In order to avoid codeswith distance equal to zero, throughout the rest of the section we only considerelements β ∈ F ∗ q n \ F ∗ q m . Proposition 4.20.
Let F = ( F , . . . , F r ) be a flag on F q n with the subfield F q m as its best friend and β ∈ F ∗ q n . If the code Orb β ( F ) is disjoint, then mr d f (Orb β ( F )) .Proof. Let F ′ be a flag in Orb β ( F ) with F ′ = F . As Orb β ( F ) is a disjoint flagcode, we have that F i = F ′ i for every i r . Hence, by means of Proposition2.2, for every i r , we have that d S ( F i , F ′ i ) > m . We conclude that d f ( F , F ′ ) > mr , for every F ′ ∈ Orb β ( F ) \ {F } , and the result holds.As shown in Proposition 4.1, the cardinality of a β -cyclic flag code Orb β ( F ) with F ∗ q m as its best friend is completely determined. Moreover, we know that h β i = h α l i , for the divisor l of q n − such that | β | = q n − l . Similarly, F ∗ q m = h α qn − qm − i . Moreover, it holds Stab β ( F ) = h α lcm( l, qn − qm − ) i and | Stab β ( F ) | = q n − l, q n − q m − ) . As a result, | Orb β ( F ) | = lcm (cid:16) l, q n − q m − (cid:17) l . (13)Using this notation, the next result follows. Theorem 4.21.
Let F be a flag on F q n with best friend F q m . Take β ∈ F ∗ q n andwrite h β i = h α l i with l a divisor of q n − . If Orb β ( F ) is an optimum distanceflag code and t is a dimension in the type vector of F , then m divides t and itmust hold lcm( l, q n − q m − ) l j q n − q t − k if t n, j q n − q n − t − k if t > n. Proof.
Consider a flag F on F q n with the subfield F q m as its best friend andassume that the code Orb β ( F ) is an optimum distance flag code. Hence, byapplication of Lemma 3.14, m must divide every dimension in the type vector.Moreover, by means of Theorem 3.3, all the projected codes attain the maximumpossible distance for their dimension and Orb β ( F ) is disjoint. In other words,the cardinality of every projected code coincides with | Orb β ( F ) | . In particular,this value has to satisfy the bounds for the cardinality of constant dimensioncodes of maximum distance given in Section 2 for dimensions in the type vector.As a result, if t is a dimension in the type vector, it must hold:21 yclic Orbit Flag Codes (1) If t n , then | Orb β ( F ) | j q n − q t − k and(2) if t > n , then | Orb β ( F ) | j q n − q n − t − k .Moreover, assuming h β i = h α l i for some divisor l of q n − , by using (13), theresult holds. Remark 4.22.
Observe that a dimension t satisfies the necessary conditionprovided in Theorem 4.21 if, and only if, the dimension n − t does it as well.This is due to the fact that the upper bound for the cardinality of constant di-mension codes with maximum distance of dimensions t and n − t of F q n coincide.Moreover, these upper bounds decrease as dimensions get closer to n/ . Hence,central dimensions are allowed for a smaller number elements β ∈ F ∗ q n than theother ones. In contrast, extreme dimensions, that is, m and n − m , are allowedfor every subgroup of F ∗ q n . In fact, when the acting group is F ∗ q n , we can derivethe following corollary. Corollary 4.23.
Assume that the cyclic orbit code
Orb( F ) is an optimum dis-tance flag code on F q n with the subfield F q m as its best friend. Then one of thefollowing statements holds:(1) Orb( F ) is a constant dimension code of dimension either m or n − m .(2) Orb( F ) has type vector ( m, n − m ) .In any of the three cases above, the code Orb( F ) has the largest possible size,that is, q n − q m − .Proof. This result follows by application of Theorem 3.3 when β is a primitiveelement of F ∗ q n . In this case, the cardinality of every projected code is q n − q m − .Moreover, if t is a dimension in the type vector, it has to be a multiple of m . Observe that, both m and n − m satisfy the necessary condition givenin Theorem 4.21. On the other hand, this condition is violated by any othermultiple of m . Hence, only dimensions m or n − m could appear in the typevector of F . As a result, optimum distance cyclic orbit flag codes with F q m astheir best friend could only be constructed for type vectors equal to ( m ) , ( n − m ) or ( m, n − m ) . For these three type vectors, the cardinality of Orb( F ) , which isalso q n − q m − , coincides with the largest possible size of constant dimension codeswith maximum distance for both dimensions m and n − m . Hence, it is the bestsize for optimum distance flag codes with any of these type vectors.Apart from the case where the type vector is ( m, n − m ) , we see that optimumdistance cyclic orbit flag codes with F q m as their best friend are actually cyclicorbit (subspace) codes of dimension either m or n − m . In case the dimensionis m , the code Orb( F ) is, in addition, the m -spread Orb( F q m ) of F q n .From Theorem 4.21 and Corollary 4.23, one can deduce that not every typevector is compatible with attaining the maximum possible distance once we havefixed the best friend of the generating flag of a β -cyclic orbit flag code. Thefollowing examples exhibit this fact. 22 yclic Orbit Flag Codes Example 4.24.
Let F be a flag on F with the subfield F as its best friend.This condition implies that the dimensions in the type vector of F must be evenintegers. Notice that | F ∗ | = 2 − · and h α i is the onlysubgroup of F ∗ of order . On the other hand, we have F ∗ = h α i . Since lcm(15 , , we have | Orb β ( F ) | = = 91 , for every β ∈ h α i .Now, assume that Orb β ( F ) is an optimum distance flag code. If we compare itssize with the upper bounds for the cardinality of constant dimension codes of F with maximum distance, we conclude that the dimension cannot appearin the type vector of Orb β ( F ) since − − = 65 < . In contrast, dimensions , , and satisfy the necessary condition given in Theorem 4.21. Example 4.25.
Consider a flag F on F q n with the subfield F q m as its bestfriend and let α denote a primitive element of F q n . The tables below illustratewhich dimensions are susceptible to appear in the type vector of the optimumdistance β -cyclic orbit flag code generated by F for different choices of β andspecific values of q, n and m . β | β | h β i ∩ F ∗ q m | Orb β ( F ) | Allowed dimensions Max. distance α F ∗ α F ∗ α F ∗
820 1, 2, 6, 7 12 α F ∗
656 1, 2, 6, 7 12 α F ∗
410 1, 2, 6, 7 12 α F ∗
328 1, 2, 6, 7 12 α F ∗
205 1, 2, 3, 5, 6, 7 24 α F ∗
164 1, 2, 3, 5, 6, 7 24 α { }
205 1, 2, 3, 5, 6, 7 24 α F ∗
82 1, 2, 3, 4, 5, 6, 7 32 α F ∗
80 1, 2, 3, 4, 5, 6, 7 32 α F ∗
41 1, 2, 3, 4, 5, 6, 7 32 α F ∗
40 1, 2, 3, 4, 5, 6, 7 32 α { }
41 1, 2, 3, 4, 5, 6, 7 32 α F ∗
20 1, 2, 3, 4, 5, 6, 7 32 α F ∗
16 1, 2, 3, 4, 5, 6, 7 32 α F ∗
10 1, 2, 3, 4, 5, 6, 7 32 α F ∗ α F ∗ α F ∗ α { } α F ∗ α F ∗ { } Table 2: Values for q = 3 , n = 8 , m = 1 and all subgroups of F ∗ .As it occurs when considering Galois β -cyclic flag codes, in these tableswe can see that different subgroups of F ∗ q n (hence, subgroups with differentorder) can provide the same β -cyclic orbit flag code. Furthermore, there aredifferent subgroups providing in turn different orbits but sharing the set ofallowed dimensions and, as a consequence, also sharing the maximum possiblevalue for the distance. For instance, in Table 3, both subgroups h α i and h α i yclic Orbit Flag Codes give us the same orbit. On the other hand, the orbits under the action of h α i and h α i have different cardinality (thus, they are different codes) but their setsof allowed dimensions are equal. β | β | h β i ∩ F ∗ q m | Orb β ( F ) | Allowed dimensions Max. distance α F ∗ α F ∗
455 2, 10 8 α F ∗
273 2, 4, 8, 10 24 α F ∗
195 2, 4, 8, 10 24 α { }
455 2, 10 8 α F ∗
105 2, 4, 8, 10 24 α F ∗
91 2, 4, 8, 10 24 α F ∗
65 2, 4, 6, 8, 10 36 α F ∗
39 2, 4, 6, 8, 10 36 α F ∗
35 2, 4, 6, 8, 10 36 α { }
91 2, 4, 8, 10 24 α { }
65 2, 4, 6, 8, 10 36 α F ∗
21 2, 4, 6, 8, 10 36 α F ∗
15 2, 4, 6, 8, 10 36 α F ∗
13 2, 4, 6, 8, 10 36 α { }
35 2, 4, 6, 8, 10 36 α F ∗ α F ∗ α { }
13 2, 4, 6, 8, 10 36 α F ∗ α { } α { } α F ∗ { } Table 3: Values for q = 2 , n = 12 , m = 2 and all subgroups of F ∗ . Remark 4.26.
Observe that results 4.21 and 4.23 give us necessary conditionson the type vector for the existence of optimum distance β -cyclic orbit flagcodes but the problem of constucting them remains open. In Subsection 4.1we have characterized optimum distance Galois β -cyclic flag codes and builtthem by providing a suitable subgroup h β i of F ∗ q n . Recall that in that case,the allowed dimensions correspond to the divisors appearing in the type vectorof the generating Galois flag. Looking at Table 3, for instance, we can obtainoptimum distance Galois β -cyclic flag codes of types (2 , and (2 , .Apart from optimum distance cyclic flag codes of Galois type, as far as weknow, there are only two constructions of optimum distance flag codes givenby the action of a cyclic subgroup of F q n . One of them can be found in [14,Prop. 2.5], where the author, for every prime power q , provide a cyclic orbit fullflag code on F q (hence, of type (1 , ) with maximum distance as a matchingcode obtained from the action of F ∗ q . The same argument allows us to buildoptimum distance cyclic orbit flag codes with best friend F q of type (1 , n − asmatching codes for every n > . On the other hand, in [2], the authors presenta construction of an optimum distance orbit full flag code on F q k arising from24 yclic Orbit Flag Codes the action of a subgroup of GL(2 k, q ) that is a cyclic group generated by thecompanion matrix of a primitive polynomial of degree k in F q [ x ] . Observe thatthis action can be naturally translated into our scenario by identifying such acompanion matrix with a primitive element of F q k , as it was pointed out in [23,Lemma 21]. We have introduced the concept of cyclic orbit flag code as a generalization ofcyclic orbit (subspace) code to the flag codes setting. Following the viewpointof [10], we analyze the structure and properties of this family of codes by defin-ing the best friend of a flag. This approach allows us to easily compute thecardinality of the code and to provide bounds for its distance.In particular, we explore families of codes attaining the extreme possiblevalues for the distance. For the minimum one, we introduce the family of Galoiscyclic flag codes, whose elements present a rich structure of nested spreadscompatible with the action of F ∗ q n on flags. We also study the subcodes ofGalois cyclic flag codes whose structure is also orbital cyclic, the Galois β -cyclic flag codes, and show that we can improve the distance of such codes bychoosing a suitable β to attain even the maximum possible one. On the otherhand, concerning optimum distance flag codes with a fixed best friend, we haveprovided a necessary condition on the type vector of orbit flag codes that attainthe maximum possible distance and arise also from the action of subgroups of F ∗ q n .In future work we want to come up with other constructions of β -cyclicorbit flag codes as well as to study conditions and properties of cyclic obit codeswith a prescribed distance not necessarily being the maximum one. Despite thestudy of union of cyclic and β -cyclic orbit flag codes has not been addressed inthis paper, it would be also interesting to tackle this problem. In addition, wewould like to exploit the structure of cyclic orbit flag codes in order to determineefficient decoding algorithms taking advantage of the ones already designed forcyclic orbit (subspace) codes in [23]. References [1] R. Ahlswede, N. Cai, R. Li and R. W. Yeung,
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