Partial Spreads in Random Network Coding
aa r X i v : . [ c s . I T ] J un PARTIAL SPREADS IN RANDOM NETWORK CODING
ELISA GORLA AND ALBERTO RAVAGNANI Institut de Mathématiques , Université de NeuchâtelRue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland A BSTRACT . Following the approach by R. Kötter and F. R. Kschischang, we study network codesas families of k -dimensional linear subspaces of a vector space F nq , q being a prime power and F q the finite field with q elements. In particular, following an idea in finite projective geometry,we introduce a class of network codes which we call partial spread codes . Partial spread codesnaturally generalize spread codes. In this paper we provide an easy description of such codes interms of matrices, discuss their maximality, and provide an efficient decoding algorithm.
0. I
NTRODUCTION
The topology of a network is well-modeled by a directed multigraph. Vertices without incom-ing edges play the role of sources and vertices without outgoing edges play the role of sinks .Vertices which are neither sources nor sinks are called nodes . The interest in network model-ing is due to its several applications in technology (distributed storage, peer-to-peer networkingand, in particular, wireless communications).In [1] Ahlswede, Cai, Li, and Yeung discovered that the information rate may be improvedby employing coding at the nodes of a network (instead of simply routing). Moreover, Li, Caiand Yeung proved in [14] that, in a multicasting situation, maximal information rate can beachieved by allowing the nodes to transmit linear combinations of the inputs they receive,provided that the size of the base field is large enough.A turning point in the study of linear network codes was the paper [12] by R. Kötter and F.R. Kschischang. The authors suggested an algebraic approach to the topic, developing a clearand rigorous mathematical setup. Interesting connections with classical projective geometryalso emerged. Several other interesting papers followed the same approach, e.g., [6], [7], and[13].In this paper, we propose and study a class of network codes, which fit within the sameframework. In Section 1 the algebraic approach by Kötter and Kschischang is briefly recalled.In Section 2 we introduce a family of network codes which we call partial spread codes, andwhich generalize spread codes (see [16]). Our codes have the same cardinality and distancedistribution as the codes proposed in [8]. The elements of our codes however are given asrowspaces of appropriate matrices in block form. The structure of this family of matrices allowus to derive properties of the code, which we discuss in Section 3. In particular, we establishthe maximality of partial spread codes with respect to containment. Based on the same blockmatrix structure, in Section 4 we are able to give an efficient decoding algorithm.
E-mail address : [email protected], [email protected] .2010 Mathematics Subject Classification.
Key words and phrases. network code, spread code, subspace distance.
1. P
RELIMINARIES
Let q be a prime power and let F q denote the finite field with q elements. Fix an integer n > P ( F nq ) be the projective geometry of F nq , i.e., the set of all the vector subspaces of F nq .Following [12], a q -ary network code of length n is defined to be a subset C ⊆ P ( F nq ) with atleast two elements. The subspace distance on P ( F nq ) is the distance map d : P ( F nq ) × P ( F nq ) → N defined, for any U , V ∈ P ( F nq ), by d ( U , V ) : = dim( U ) + dim( V ) − U ∩ V ).As in classical Coding Theory, the minimum distance of a network-code C ⊆ P ( F nq ) is theinteger d ( C ) : = min { d ( U , V ) : U , V ∈ C , U V } . The maximum dimension of C is denoted anddefined by ℓ ( C ) : = max V ∈ C dim( V ). Let us briefly recall from [12] the framework for errorsand erasures in random network coding. If 1 ≤ e < n is an integer, then an e - erasure on anelement V ∈ P ( F nq ) such that dim( V ) ≥ e is the projection of V onto an e -dimensional subspaceof V . In other words, an e -erasure replaces V with an e -dimensional subspace of V . A t -dimensional error E on an element V ∈ P ( F nq ) corresponds to the direct sum V ⊕ E , where E ∈ P ( F nq ), dim( E ) = t and E ∩ V = { } . If C ⊆ P ( F nq ) is a network code, then an input codeword V ∈ C and its output U ∈ P ( F nq ) are related by U = H e ( V ) ⊕ E , where 1 ≤ e ≤ dim( V ), H e is an e -erasure operator and E ∈ P ( F nq ) the error. As usual, one can bound the number of erasures anderrors that can take place such that a minimum distance decoder is guaranteed to successfullyreturn the sent codeword. Theorem 1 ([12], Theorem 2) . Let C ⊆ P ( F nq ) be a network-code of minimum distance d . As-sume that an input V ∈ C and its output U ∈ P ( F nq ) are related by U = H e ( V ) ⊕ E , where e ≤ ℓ ( C ), H e is an e -erasure and E ∈ P ( F nq ) is an error. Set t : = dim( E ). A minimum distancedecoder corrects U in V , provided that 2( t + ℓ ( C ) − e ) < d .A natural class of network codes is obtained by considering subsets of P ( F nq ), all of whoseelements have the same dimension 1 ≤ k ≤ n −
1. Such codes are called constant dimension codes. By introducing the Grassmannian variety G q ( k , n ) : = { V ∈ P ( F nq ) : dim( V ) = k } ,a q -ary constant dimension network code of lenght n and dimension k is simply a subset C ⊆ G q ( k , n ) of at least two elements. It easily follows from the definition that any constantdimension network code has even minimum distance. Remark 2.
The cardinality of the Grassmannian variety G q ( k , n ) is known to be · nk ¸ q : = ( q n − q n − − · · · ( q n − k + − q k − q k − − · · · ( q − = k − Y i = q n − i − q k − i − Theorem 3 (Singleton-like Bound, [12], Theorem 9) . Let C ⊆ G q ( k , n ) be a network code ofminimum distance d . Then | C | ≤ · n − ( d − { k , n − k } ¸ q .The family of spread codes has been introduced in [16], and an efficient decoding algorithmfor such codes has been provided in [9]. Definition 4. A k - spread of F nq is a collection of subspaces { V i } ti = of F nq (here we take k < n )such that ARTIAL SPREADS IN RANDOM NETWORK CODING 3 (1) dim V i = dim V j = k for any i , j ∈ {
1, ..., t } ,(2) V i ∩ V j = { } whenever i j ,(3) F nq = S ti = V i . Remark 5. A k -spread of F nq exists if and only if k divides n (see [11], Corollary 4.17). Fromthe definition we see that if { V i } ti = is a k -spread of F nq then t = ( q n − q k − G q ( k , n ), a k -spread in F nq is a q -ary network code of lenght n , dimension k and minimum distance 2 k . It is easily checked that spread codes meet the Singleton-like bound(Theorem 3). 2. P ARTIAL SPREAD CODES
In this section we introduce a generalization of the definition of spread and a related familyof network codes, whose parameters k and n can be chosen freely. Definition 6. A partial k -spread of F nq is a subset C ⊆ G q ( k , n ) such that U ∩ V = { } for any U , V ∈ C with U V . A partial k -spread of F nq with at least two elements is a q -ary networkcode of lenght n , dimension k and minimum distance 2 k . We will call such a code a partialspread code . Lemma 7.
Let C ⊆ G q ( k , n ) be a partial spread code. Denote by r the remainder obtaineddividing n by k . Then | C | ≤ q n − q r q k − Proof.
Since C is a set of k -dimensional vector subspaces of F nq with trivial pairwise intersec-tions, we deduce | C | · ( q k − + ≤ q n . Since k divides n − r , ( q n − r − q k −
1) is an integer.Hence | C | ≤ ¹ q n − q k − º = ¹ q r ( q n − r − q k − + q r − q k − º = q n − q r q k − (cid:3) The bound given in Lemma 7 admits some non-trivial improvements. See [3] and [4] fordetails. The following lower bound for partial k -spread in F nq is due to A. Beutelspacher (see [2]for a non-constructive proof). Lemma 8.
Let q be a prime power and let 1 ≤ k < n be integers. Write n = hk + r with 0 ≤ r ≤ k −
1. Denote by A q ( k , n , 2 k ) the largest possible size of a network code C ⊆ G q ( k , n ) of minimumdistance 2 k . Then A q ( n , k , 2 k ) ≥ ( q n − q r )/( q k − − q r + Remark 9.
An alternative proof of Lemma 8 is given in [8], Theorem 11. For interestingdiscussions on the sharpness of the bound see [5] and [10].Here we introduce a construction for partial spread codes whose size attains the lower boundof Lemma 8. Notice that the vector spaces of the partial spread are given as row spaces ofappropriate easy-computable matrices.
Lemma 10 ([15], Ch. 2.5) . Let q be a prime power and let F q be the finite field with q elements.Choose an irreducible monic polynomial p ∈ F q [ x ] of degree k ≥ p = P ki = p i x i . Define PARTIAL SPREADS IN RANDOM NETWORK CODING the companion matrix of p by M ( p ) : = · · ·
00 0 1 0... . . . ...0 0 0 1 − p − p − p · · · − p k − .The F q -algebra F q [ P ] is a finite field with q k elements. Notation 11.
Let V be a vector space over a field F and let S ⊆ V be any subset. The vectorspace generated by S , i.e., the smallest vector subspace of V containing S , is denoted by 〈 S 〉 .We always have dim F 〈 S 〉 ≤ | S | . Lemma 12.
Let V be a finite-dimensional vector space over a field F . Let D ⊆ V be any subsetand set d : = dim F 〈 D 〉 . Choose a finite subset S ⊆ D . Then dim F 〈 D \ S 〉 ≥ d − | S | . Proof.
Since D = ( D \ S ) ∪ S we have 〈 D \ S 〉 + 〈 S 〉 ⊇ 〈 ( D \ S ) ∪ S 〉 = 〈 D 〉 . It followsdim F 〈 D \ S 〉 + dim F 〈 S 〉 ≥ d + dim F 〈 D \ S 〉 ∩ 〈 S 〉 .Since dim F 〈 S 〉 ≤ | S | we conclude dim F 〈 D \ S 〉 + | S | ≥ d . (cid:3) Theorem 13.
Let q be a prime power and let F q be the finite field with q elements. Chooseintegers 1 ≤ k < n and write n = hk + r with 0 ≤ r ≤ k −
1. Assume h ≥
2. Let p , p ′ ∈ F q [ x ] be twoirreducible monic polynomials of degree k and k + r respectively, and let P : = M ( p ), P ′ : = M ( p ′ )be their companion matrices. For any 1 ≤ i ≤ h − M i ( p , p ′ ) : = ©£ k · · · k I k A i + · · · A h − A ( k ) ¤ : A i + , ..., A h − ∈ F q [ P ], A ∈ F q [ P ′ ] ª ,where 0 k is the k × k matrix with zero entries, I k is the k × k identity matrix, and A ( k ) denotesthe last k rows of A . The set C : = h − [ i = © rowsp( M ) : M ∈ M i ( p , p ′ ) ª ∪ © rowsp £ k · · · k k × r I k ¤ª is a partial spread code in F nq of dimension k . In particular, the minimum distance of C is 2 k . Proof.
Choose matrices M M ∈ M i ( p , p ′ ) and set V : = rowsp( M ), V : = rowsp( M ). Sinceby definition d ( V , V ) = k − V ∩ V ), we have d ( V , V ) = k if and only ifrk · M M ¸ = k .Since M M , it is possible to find either in · M M ¸ , or in · M M ¸ , a submatrix in one of thefollowing three forms: N : = · I k B k I k ¸ , N : = · I k B I k B ¸ , N : = · I k X ( k ) I k Y ( k ) ¸ ,with B , B B ∈ F q [ P ] and X Y ∈ F q [ ˜ P ]. Let us compute the ranks of such matrices case bycase. The rank of N is easily computed asdim F q rowsp £ I k B ¤ + dim F q rowsp £ k I k ¤ − dim F q ¡ rowsp £ I k B ¤ ∩ rowsp £ k I k ¤¢ = k .The rank of N is equal to the rank of · I k B k B − B ¸ . ARTIAL SPREADS IN RANDOM NETWORK CODING 5
Since B B , by Lemma 10 we get that B − B is an invertible matrix, hencedet · I k B k B − B ¸ = det( B − B ) N ) = k . In order to study the latter case, consider the 2( k + r ) × k + r )matrix H : = · I k + r XI k + r Y ¸ .By using the same argument as above, we get rk( H ) = k + r ). Delete from H the rows fromone to r and from k + r + k + r . A matrix of size 2 k × (2 k + r ), say ˜ H , is obtained. Weobserve that the rows of ˜ H are exactly the rows of N with r extra zeroes in the beginning. Inparticular, rk( ˜ H ) = rk( N ). By Lemma 12 we get rk( ˜ H ) ≥ k + r ) − r = k and so rk( N ) = k .To conclude the proof, take a matrix M ∈ M i ( p , p ′ ) and set M : = £ k · · · k k × r I k ¤ . Itfollows rk · M M ¸ = k .These arguments prove that that C is a set of k -dimensional vector subspaces of F nq , whosepairwise intersections are trivial. (cid:3) Notation 14.
The partial spread code C defined in the statement of Theorem 13 will be denotedby C q ( k , n ; p , p ′ ). Since, for any code C ⊆ G q ( k , n ), C ⊥ ⊆ G q ( n − k , n ) is a code with the samecardinality and the same distance distribution as C (see [12], Section III), we always assume1 ≤ k ≤ n /2. Remark 15.
Partial spread codes provide a generalization of spread codes (see [16], Definition2). Indeed, it is easily seen that spread codes are obtained by taking r : = p ′ : = p in thestatement of Theorem 13. On the other hand, partial spread codes exist also when k does notdivide n . Example 16.
Here we construct a partial spread code of lenght 7 and dimension 2 over thebinary field F . Let ( q , k , n ) : = (2, 2, 7) and observe that n ≡ k . Hence, in the notation ofTheorem 13, r =
1. Take irreducible monic polynomials p : = x + x + p ′ : = x + x + ∈ F [ x ] ofdegree k and k + r , respectively. The companion matrices of p and p ′ are easily computed asfollows: P : = M ( p ) = · ¸ , P ′ : = M ( p ′ ) = .As a consequence, the elements of C (2, 7; p , p ′ ) are the row spaces of all the matrices in thefollowing forms: · A A (2) ¸ , · B (2) ¸ , · ¸ ,where A is any matrix in F q [ P ] and A (2) , B (2) denote the last two rows of any A , B ∈ F q [ P ′ ]. Itcan be checked that C (2, 7; p , p ′ ) has 2 · + + =
41 elements. The cardinality computationwill be easily generalized in Proposition 17.
PARTIAL SPREADS IN RANDOM NETWORK CODING
3. S
OME PROPERTIES OF PARTIAL SPREAD CODES
In this section we discuss some relevant properties of partial spread codes introduced inTheorem 13. In particular, Proposition 17 provides their size and Proposition 20 proves theirmaximality, with respect to inclusion, as collections of k -dimensional vector subspaces of F nq with trivial pairwise intersections. Proposition 17.
Let C : = C q ( k , n ; p , p ′ ) be a partial spread code. The size of C is given by theformula | C | = q n − q r q k − − q r + Proof.
We follow the notation of Theorem 13. Let X , Y be matrices in F q [ P ′ ] and assume X ( k ) = Y ( k ) . If X Y we have rk · I k + r XI k + r Y ¸ = k + r )and so, as in the proof of Theorem 13, rk · I k X ( k ) I k Y ( k ) ¸ = k ,which yields a contradiction. It follows that X = Y . Notice that the matrices in the statement ofTheorem 13 are given in row-reduced echelon form, which is canonical (see [17], Chapter 2.2).As a consequence, the size of C is easily computed as | C | = + q k + r h − X i = q ki = ( q n − q r )/( q k − − q r + (cid:3) Corollary 18.
Let C : = C q ( k , n ; p , p ′ ) be a partial spread code. Denote by A q ( k , n , 2 k ) thelargest possible size of a network code in G q ( k , n ) of minimum distance 2 k . Let r be the remain-der obtained dividing n by k . Then A q ( k , n , 2 k ) − | C | ≤ q r − Proof.
Combine Lemma 7 and Proposition 17. (cid:3)
Remark 19.
In [8] T. Etzion and A. Vardy provide a construction of partial spread codes (seethe proof of Theorem 11). Their codes have the same cardinality and minimum distance as C q ( k , n ; p , p ′ ). The main contribution of this paper is introducing a block-matrices descriptionof partial spread codes. Thanks to our constrution, in Section 5 we are able to provide anefficient decoding algorithm for partial spread codes. In the next proposition, we discuss themaximality of partial spread codes. Proposition 20.
Let N q ( k , n , 2 k ) be the set of all the possible network codes C ⊆ G q ( k , n ) ofminimum distance 2 k . Let C : = C q ( k , n ; p , p ′ ) be a partial spread code. Then C is a maximalelement of N q ( k , n , 2 k ) with respect to inclusion. Proof.
We must prove that there is no partial k -spread C ′ in F nq such that C ′ ⊇ C and | C ′ | > | C | .Write n = hk + r with 0 ≤ r < k and h ≥ k -spread C : = C \ © rowsp £ k · · · k k × r I k ¤ª .Assume, by contradiction, that there exists a partial k -spread C ′ in F nq such that C ′ ⊇ C and | C ′ | ≥ | C |+
2. Set S : = S C \{ } . By combining Theorem 13 and Proposition 17 we easily compute | C | = ( q n − q r )/( q k − − q r , | S | = ( q k − · | C | = q n − q k + r . ARTIAL SPREADS IN RANDOM NETWORK CODING 7
The set X : = { x ∈ F nq : x i = i =
1, ..., ( h − k } is a vector subspace of F nq of dimension k + r .We clearly have an inclusion X ⊆ F nq \ S . Since | F nq \ S | = q n − ( q n − q k + r ) = q k + r ,we deduce X = F nq \ S , F nq = X ⊔ S , with X a ( k + r )-dimensional vector subspace of F nq . Since C ′ ⊇ C ⊇ C , | C ′ | ≥ C + s ∈ S there exists a V s ∈ C such that s ∈ V s , we deducethe existence of two k -dimensional vector subspaces V , V ∈ C ′ such that V ∩ V = { } and V , V ⊆ X . Since X is a vector subspace of F nq containing V ∪ V and, by definition, V + V is the smallest vector subspace of F nq containing both V and V , we conclude V + V ⊆ X . Itfollows dim( V ) + dim( V ) − dim( V ∩ V ) ≤ dim( X )and so 2 k ≤ k + r , which is a contradiction. (cid:3) Remark 21.
Proposition 20 ensures that a partial spread code C q ( k , n ; p , p ′ ) cannot be im-proved (as a network code in G q ( k , n ) of minimum distance 2 k ) by adding new codewords.4. T HE BLOCK STRUCTURE
Here we investigate the block structure of partial spread codes introduced in the statementof Theorem 13. This will allow us to produce an efficient decoding algorithm, which we presentin the next section. The results of this section are a generalization of those contained in [9].
Lemma 22.
Let C : = C q ( k , n ; p , p ′ ) be a partial spread code and let V ∈ C be a codeword, say V : = rowsp £ S · · · S h − S ¤ ,where the S i ’s are k × k matrices and S is a k × ( k + r ) matrix. Let X ⊆ F nq be a t -dimensionalvector subspace given as the row space of a matrix of the form £ M · · · M h − M ¤ ,where the M i ’s are k × k matrices and M is a k × ( k + r ) matrix . If d ( V , X ) < k then X decodesto V . Moreover, for any 1 ≤ i ≤ h − S i = k ,(2) rk( M i ) ≤ ( t − Proof.
Since the minimum distance of C is 2 k (Theorem 13) and d ( V , X ) < k , the space X obviously decodes to V . Let us prove (1) ⇒ (2) . Without loss of generality, we assume that £ S · · · S h − S ¤ is in row-reduced echelon form. Assume that for a fixed index 1 ≤ i ≤ h − S i = k . Since d ( V , X ) < k , we have dim F q ( V ∩ X ) > t /2. By definition of C , exactly oneof the following cases occurs:(a) there exists an index 1 ≤ j ≤ h − j i such that S j = I k ;(b) S j = k for any 1 ≤ j ≤ h − M i j defined by M i j : = · k I k M i M j ¸ . Notice that t ≤ k . This assumption is not restrictive from the following point of view: the decoder can stopcollecting incoming vectors as soon as it receives k inputs (as an alternative, k linearly independent inputs); then itcan attempt to decode the collected data. PARTIAL SPREADS IN RANDOM NETWORK CODING
We get rk( M i j ) ≤ dim( V + X ) = k + t − dim F q ( V ∩ X ) < k + t /2. Assume by contradiction thatrk( M i ) > ( t − k coloumns of M i j (which are linearly independent anddo not lie in the space generated by the first k ) we easily deduce the following contradiction: k + ( t − < rk · k I k M i M j ¸ < k + t /2.In the latter case, by definition of C , we have V = rowsp £ k · · · k k × r I k ¤ . Hence k + ( t − < rk · k k × r I k M i M ¸ ≤ dim( V + X ) = k + t − dim F q ( V ∩ X ) < k + t /2,a contradiction. Now we prove (2) ⇒ (1) . Assume rk( M i ) ≤ ( t − ≤ i ≤ h − S i k then, by definition of C , rk( S i ) = k . Denote by π : F nq → F kq the projection on thecoordinates ki + ki +
2, ..., k ( i + S i ) = π ( V ) and rk( S i ) = k , we get that π | V issurjective. Since dim F q ( V ) = k , it follows that π | V is also injective. As a consequence,dim F q ( V ∩ X ) = dim F q π ( V ∩ X ) ≤ dim F q ( π ( V ) ∩ π ( X )) ≤ dim F q π ( X ) = rk( M i ) ≤ ( t − d ( V , X ) < k . (cid:3) Remark 23.
Lemma 22 has the following useful interpretation. Assume that a partial spreadcode C : = C q ( k , n ; p , p ′ ) is used for random network coding and a t -dimensional vector space X : = rowsp £ M · · · M h − M ¤ is received. Assume the existence of a (unique) codeword V ∈ C such that d ( V , X ) < k (i.e., X decodes to V ). If rk( M i ) ≤ ( t − ≤ i ≤ h − V = rowsp £ k · · · k k × r I k ¤ . Otherwise, let i denote the smallest integer 1 ≤ i ≤ h − M i ) > ( t − A i + , ..., A h − ∈ F q [ P ] and a uniquematrix A ∈ F q [ P ′ ] such that V = rowsp £ k · · · k I k A i + · · · A h − A ( k ) ¤ , where theidentity matrix I k is the i -th k × k block. Lemma 24.
With the setup of Remark 23, assume that V rowsp £ k · · · k k × r I k ¤ . Forany i + ≤ j ≤ h − d ¡ rowsp £ I k A j ¤ , rowsp £ M i M j ¤¢ < k , d ¡ rowsp £ I k A ( k ) ¤ , rowsp £ M i M ¤¢ < k . Proof.
Fix an integer j such that i + ≤ j ≤ h − π : F nq → F kq the projection on thecoordinates ki + ki +
2, ..., k ( i + k j + k j +
2, ..., k ( j + π ( V ) = rowsp £ I k A j ¤ and π ( X ) = rowsp £ M i M j ¤ . In particular, rk( π | V ) = k . As a consequence, dim F q ker( π | V ) ≤ k − k = π | V is injective. By the trivial inclusion of vector spaces π ( V ∩ X ) ⊆ π ( V ) ∩ π ( X ) it followsdim F q π ( V ∩ X ) ≤ dim F q ( π ( V ) ∩ π ( X )). Hence d ( π ( V ), π ( X )) = k + dim F q π ( X ) − F q ( π ( V ) ∩ π ( X )) ≤ k + t − F q π ( V ∩ X ) = k + t − F q ( V ∩ X ) = d ( V , X ) < k .In order to prove that d ¡ rowsp £ I k A ( k ) ¤ , rowsp £ M i M ¤¢ < k we may notice that the sameargument still works if we choose as π : F nq → F k + rq the projection on the coordinates ki + ki +
2, ..., k ( i + k ( h − + k ( h − +
2, ..., kh , kh +
1, ..., kh + r . (cid:3) Remark 25.
By Lemma 24, when decoding a partial spread code we may restrict to one ofthe two the cases n = k and n = k + r , with 1 ≤ r ≤ k −
1. Moreover, the lemma allows us toparallelize the computation, reducing the decoding complexity to the case n = k + r . ARTIAL SPREADS IN RANDOM NETWORK CODING 9
5. D
ECODING PARTIAL SPREAD CODES
In [12] R. Kötter and F. R. Kschischang illustrate a general network code construction and arelated efficient algorithm to decode them. A more efficient algorithm to decode the same codesappears in [13]. After recalling the definition of
Reed-Solomon like code, we use the resultsestablished in the previous section to adapt any decoding algorithm for such codes to partialspread codes of the form C q ( k , n ; p , p ′ ). Definition 26.
Let q be a prime power and let n > A : = { α , ..., α k } ⊆ F q n be aset of F q -linearly independent elements. Choose an integer s ≤ k and denote by F sq n [ x ] the vectorspace of the linearized polynomial of degree at most s and coefficients in F q (see [12], Section5.A, for details). Fix an F q -isomorphism of vector spaces ϕ : F q n → F nq . The Reed-Solomon like code associated to the 6-tuple ( q , n , k , s , A , ϕ ) is the set KK q ( n , k , s , A , ϕ ) : = rowsp ϕ ( f ( α )) ϕ ( f ( α )) I k ... ϕ ( f ( α k − )) ϕ ( f ( α k )) : f ∈ F sq n [ x ] . Remark 27.
A Reed-Solomon like code KK q ( n , k , s , A , ϕ ) is a subset of the Grassmannian vari-ety G q ( k , k + n ). As a consequence, it is a q -ary network code of lenght k + n and dimension k .The size of such a code is given by the easy-computable formula | KK q ( n , k , s , A , ϕ ) | = q sn . See[12], Section 5.1, for a more detailed discussion. Lemma 28.
Let q be a prime power and let k ≥ p an irreducible monicpolynomial p ∈ F q [ x ] of degree k and let P : = M ( p ) be its companion matrix. Choose a root λ ∈ F q k of p . Denote by ϕ : F q n → F nq the F q -isomorphism defined, for any 0 ≤ i ≤ k −
1, by λ i e i + , where { e , ..., e k } is the canonical basis of F kq . Let A ∈ F q [ P ] and, for any 1 ≤ i ≤ k , let A i ∈ F kq denote the i -th row of A . For any 1 ≤ i ≤ k we have ϕ − ( A i ) = λ j − ϕ − ( A ). In particular,if f ∈ F q k [ x ] is defined by f ( x ) : = ϕ − ( A ) x , then A = ϕ ( f ( λ )) ϕ ( f ( λ )) ϕ ( f ( λ ))... ϕ ( f ( λ k − )) . Proof.
Use [9], Proposition 15, with n = k . (cid:3) Notation 29.
In the costruction of a partial spread code C q ( k , 2 k + r , p , p ′ ) with 0 ≤ r ≤ k − p is never involved (see Theorem 13). As a consequence, we write C q ( k , 2 k + r ; p ′ ) in this case. Remark 30.
By Lemma 25, in order to decode a partial spread code C q ( k , n ; p , p ′ ) we mayrestrict to decoding partial spread codes of the form C q ( k , 2 k + r ; p ), with 0 ≤ r ≤ k −
1. The case r = C q ( k , 2 k ; p ) \ © rowsp £ k I k ¤ª is a Reed-Solomonlike code and so we may simply proceed as in the following Algorithm 5. Remark 31.
In [9], a decoding procedure for C q ( k , hk ; p ) spread codes which is independent ofthose of [12] and [13] is proposed. Lemma 24 allows us to apply the decoding algorithm from [9]to partial spread codes. This algorithm is particularly efficient in the case k ≪ n . Algorithm 1
Decoding a C q ( k , 2 k ; p ) code. Data : a decodable t -dimensional row space, X , of a ( k × k )-matrix £ M M ¤ . Result : the unique V ∈ C q ( k , 2 k ; p ) such that d ( V , X ) < k , given as a matrix in row-reducedechelon form whose row space is V . if rk( M ) ≤ ( t − then V = rowsp £ k I k ¤ . else use a decoding algorithm for Reed-Solomon like codes on C q ( k , 2 k ; p ) \ © rowsp £ k I k ¤ª . end if Now we focus on a decoding procedure for partial spread codes of the form C q ( k , 2 k + r ; p ) with1 ≤ r ≤ k −
1. To be precise, in the following Proposition 32 we construct a canonical embedding of a partial spread code C q ( k , 2 k + r ; p ) into the spread code C q ( k + r , 2( k + r ); p ). Any decodingprocedure for C q ( k + r , 2( k + r ); p ) gives, in this way, a decoding procedure for C q ( k , 2 k + r ; p ). Proposition 32.
Let C : = C q ( k , 2 k + r , p ) be a partial spread code with 1 ≤ r ≤ k −
1. Let X : = rowsp £ M M ¤ be a t -dimensional vector space in F n + rq , where M is a ( k × k )-matrixand M is a matrix of size k × ( k + r ). Assume the existence of a matrix A ∈ F q [ P ] such that d ¡ rowsp £ I k A ( k ) ¤ , rowsp £ M M ¤¢ < k . Define the following two ( k + r ) × ( k + r )-matrices: M : = · r r × k k × r M ¸ , M : = · r × ( k + r ) M ¸ .We have d ¡ rowsp £ I k + r A ¤ , rowsp £ M M ¤¢ < k + r . Proof.
Set V : = rowsp £ I k A ( k ) ¤ and observe that the hypothesis d ( V , X ) < k can be restatedas dim( V ∩ X ) > t /2. Define V : = rowsp £ I k + r A ¤ and X : = rowsp £ M M ¤ . By construction,dim F q X = dim F q X = t and dim F q ( V ∩ X ) ≥ dim F q ( V ∩ X ). It follows d ( V , X ) = dim F q V + dim F q X − F q ( V ∩ X ) = k + r + t − F q ( V ∩ X ) ≤ k + r + t − F q ( V ∩ X ) < k + r + t − t /2) = k + r ,as claimed. (cid:3) Remark 33.
Proposition 32 has the following useful interpretation. Assume that a partialspread code C q ( k , 2 k + r ; p ) is given, with 1 ≤ r ≤ k −
1, and X : = rowsp £ M M ¤ is received( M and M being as in the statement of the proposition). Then we may construct the matrices M and M as described and consider the vector space X : = rowsp £ M M ¤ . The minimumdistance of the (partial) spread code C q ( k + r , 2( k + r ); p ) is 2( k + r ). By Proposition 32, if X decodes to V : = rowsp £ I k A ( k ) ¤ in C q ( k , 2 k + r ; p ), then X decodes to V : = rowsp £ I k + r A ¤ in C q ( k + r , 2( k + r ); p ). It follows that Algorithm 5 (with k ← k + r ) applied to X produces £ I k + r A ¤ .Finally, V is the rowspace of the matrix obtined by deleting the first r rows and the first r coloumns of £ I k + r A ¤ . This discussion leads to the following Algorithm 5. Remark 34.
By Proposition 32, in Algorithm 5 we may replace the use of Algorithm 5 withany other decoding algorithm for spread codes.
ARTIAL SPREADS IN RANDOM NETWORK CODING 11
Algorithm 2
Decoding a C q ( k , 2 k + r ; p ) code with 1 ≤ r ≤ k − Data : a decodable t -dimensional row space, X , of a ( k × k + r )-matrix £ M M ¤ . Result : the unique V ∈ C q ( k , 2 k + r ; p ) such that d ( V , X ) < k , given as a matrix in row-reducedechelon form whose row space is V . if rk( M ) ≤ ( t − then V = rowsp £ k k × r I k ¤ . else construct the matrix £ M M ¤ as explained in Lemma 32. Then use Algorithm 5 with C q ( k + r , 2( k + r ); p ) on £ M M ¤ . Delete the first r rows and the first r coloumns of theoutput. end if C ONCLUSIONS
In this paper we provide an easy description of partial spreads over finite fields, whose in-terest dates back to classical problems in projective geometry. We suggest the use of partialspreads as network codes, investigating the mathematical properties due to our construction,proving their maximality, and providing a decoding algorithm for them.A
CKNOWLEDGMENT
The authors would like to thank Leo Storme for useful discussions on partial spreads infinite projective geometry. R
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