Construction of Error-Correcting Codes for Random Network Coding

Construction of Error-Correcting Co des for Random Net w ork Co ding T uvi Etzion Department of C omput er Science T echnion, Haifa 32 000, Israel e-mail: etzion@cs. technion.ac.il phone: 04-8294311 , fax: 04-8293900 Natalia Silber stein Department of Computer Science T echnion, Haifa 32000, Israel e-mail: natalys@cs .technion.ac.il phone: 04-8294952 , fax: 04-8293900 Ab str act — In this wor k we present error -correcting codes for r andom net work coding based on rank- metric co des, F errers diagrams, and puncturing. F or most parameters, the constructed codes are larger than all previously known code s. Classification: Information Theor y and Co ding Theor y I. Intr o duction The pr oje ctive sp ac e of order n o ver finite field F q = GF ( q ), denoted b y P q ( n ) , is the set of all subspaces of the v ector space F n q . P q ( n ) is a met ric space with th e distance function d S ( U, V ) = dim( U ) + dim( V ) − 2dim ( U ∩ V ), for all U, V ∈ P q ( n ). A code in the pro jective space is a sub s et of P q ( n ). Koett er and Ksc hischang [4 ] sho wed th at cod es in P q ( n ) are useful for correcting errors and erasures in rand om netw ork codin g. If the dimension of eac h co dew ord is a giv en integer k ≤ n th en the code forms a subset of a the Grassmannian G q ( n, k ) and called a c onstant-dimension c o de . The r ank distanc e b et we en X ,Y ∈ F m × t q is defined b y d R ( X, Y ) = rank( X − Y ). It is wel l known [2] th at th e rank distance is a metric. A co de C ⊆ F m × t q with the rank d istance is calle d a r ank-metric c o de . The connection betw een the rank- metric co des and co des in P q ( n ) was explored in [3, 4, 6]. W e represen t a k - dimensio nal subspace U ∈ P q ( n ) by a k × n matrix, in r e duc e d r ow e chelon form , whose ro ws form a basis for U . The e chelon F err ers form of a binary vector v of length n and weig ht k , E F ( v ), is a k × n matrix in reduced ro w echelon form with leading entries (of rows) in the columns indexed by th e nonzero en tries of v and ” • ” (will b e called dot ) in th e “arbitrary” entries. Example 1. L et v = 0110100 . Then E F ( v ) = 0 @ 0 1 0 • 0 • • 0 0 1 • 0 • • 0 0 0 0 1 • • 1 A . Let S b e the sub-matrix of E F ( v ) th at consists of all its columns with dots. A matrix M o ver F q is said to b e in E F ( v ) if M has the same size as S and if S i,j = 0 implies th a t M i,j = 0. E F ( v [ M ]) will b e t he matrix that result by placing the matrix M instead of S in E F ( v ). I I. Construction of Const ant Dimension Codes Let C be a constant-wei ght co de of length n , constant weigh t k , and minimum Hamming distance d H = 2 δ . Let C v b e the largest rank-metric cod e with t h e minimum distance d R = δ , such that all its codewords are in E F ( v ). N o w define co de C = ∪ v ∈ C { E F ( v [ c ] ) : c ∈ C v } . Lemma 1. F or al l v 1 , v 2 ∈ C and c i ∈ E F ( v i ) , i = 1 , 2 , d S ( E F ( v 1 [ c 1 ]) , E F ( v 2 [ c 2 ])) ≥ d H ( v 1 , v 2 ) . If d H ( v 1 , v 2 ) = 0 , then d S ( E F ( v 1 [ c 1 ]) , E F ( v 2 [ c 2 ])) = 2 d R ( c 1 , c 2 ) . Corollary 1. C ∈ G q ( n, k ) and d S ( C ) = 2 δ . Theorem 1. L et C v ⊆ F m × t q b e a r ank-metric c o de wi th d R ( C v ) = δ , such that al l its c o dewor ds ar e in E F ( v ) for some binary ve ctor v . L et S b e the sub-matrix of E F ( v ) which c or- r esp onds to th e dots p art of E F ( v ) . Then the dimension of C v is upp er b ounde d by the minimum b etwe en the numb er of dots in the last m − δ + 1 r ows of S and the numb er of dots in the first t − δ + 1 c olumns of S . Constructions for co des whic h attain the b ound of Theo- rem 1 for most imp ortan t cases are given in [1]. Examples are giv en in the follo wing table (see [5] for details): q n k d s | C | 2 6 3 4 71 2 7 3 4 289 2 8 4 4 4 573 I II. Error-Correcting Project ive Sp ace Codes Let C ∈ G q ( n, k ) with d S ( C ) = 2 δ . Let Q b e an ( n − 1)- dimensional subspace of F n q and v ∈ F n q such that v / ∈ Q . Let C ′ = C 1 ∪ C v , where C 1 = { c ∈ C : c ⊆ Q } and C v = { c ∩ Q : c ∈ C , v ∈ c } . Lemma 2. C ′ ∈ P q ( n − 1 ) and d S ( C ′ ) = 2 δ − 1 . By applying this pu ncturing metho d w ith the 7-dimensional subspace Q whose generator matrix is 0 B B B @ 1 0 . . . 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . 1 0 1 C C C A and the vector v = 10000001 , on th e co de with size 4573, and minim u m distance 4, in G 2 (8 , 4), we were able to obtain a co de with minimum distance 3 and size 573 in P 2 (7). References [1] T. Etzion and N. S il b erstein, Erro r- correcting cod es in pro jectiv e space via rank-metric co d es and F errers dia- grams, in p repara tion. [2] E. M. Gabidulin, Theory of co d es with maxim um rank dis- tance, Pr oblems on Inf or m . T r ans. , 21(1):1–12, Jan.1985. [3] M. Gadouleau and Z. Y an , On th e connection b etw een op- timal constan t- ra n k codes and optimal constant-dimension codes, 20 08, a v ailable at http://arxiv.org/abs/08 03.2262 . [4] R.Ko etter and F.R. Ksc h ischang, Co ding for errors and erasures in random netw ork co ding, IEEE T r ans. I nform. The ory , to app ear. [5] N. Silberstein, Coding theory in pro jective space, pro- p os al, 200 8, a v ailable at http://arxiv.org/abs/080 5.3528 . [6] D. Silv a, F. R. K s chisc hang, and R. Ko etter, A rank- metric app roach to error control in random netw ork co d- ing, ITW, Ber gen, Norway , pages 73–79, July 1-6, 2007.