Bounds on Covering Codes with the Rank Metric

Bounds on Cov ering Codes with the Rank Metric Maximili en Gadouleau, Stu dent Member , IEEE, and Zhiyuan Y an, Senior Member , IEEE Abstract In this paper, we in vestigate g eometrical proper ties of th e rank metric space and covering prop erties of rank metric codes. W e first establish an analy tical expression for the intersection of tw o balls with rank radii, and then deriv e a n upper bound on the volume of the u nion of multiple balls with rank radii. Using these geometrical properties, we deriv e b oth upper and lo wer bo unds on the minimum cardinality of a c ode with a gi ven rank covering radius. The g eometrical prop erties and bou nds propo sed in this paper ar e significant to the de sign, deco ding, and perf ormanc e analysis of ran k metric codes. Index T erms Error control codes, covering radiu s, r ank metric codes, geometrical proper ties, intersectio n number . I . I N T RO D U C T I O N There is a s teady strea m of works on ran k me tric co des due to their a pplications to data storag e, pub lic- key cryptosys tems, and space-time coding (se e [1], [2] and the references therein for a comprehensive literature survey), and interest in rank metric code s is strengthened by their rece nt applications in error control for random network co ding [2], decribed be low . Cons tant-dimension c odes [3] are an important class of codes for error and erasure correction in r andom linear network coding. Using the lif ting ope ration [2], rank metric codes can be readily turned into c onstant-dimens ion codes without modifying the ir This work w as supp orted in part by Thales Communications Inc. and in part by a grant from the Commonwealth of Pennsylv ania, Department of Community and Economic Develo pment, through the Pennsylvania Infrastructure T echnology Alliance (PIT A). This work was carried out while M. Gadouleau wa s at Lehigh Uni versity . Maximilien Gadouleau is wi th Crestic, Univ ersit ´ e de Reims Champagne-Arde nne, Reims, France. Zhiyuan Y an is with the Department of E lectrical and Computer Engineering, Lehigh Uni versity , Bethlehem, P A, 18015 USA (e-mail: maximilien.gadouleau @uni v-reims.fr; yan@lehigh.edu). 2 distance properties. It is shown in [3] that liftings o f rank metric cod es have many advantages compared to general c onstant-dimension codes. First, their cardinalities are optimal up to a c onstant; second, the decoding of these codes can be done in either the subspa ce metric [3] or the rank metric [2], and for both scenarios efficient d ecoding algorithms were propos ed. Despite their significance , many research problems in rank metric codes remain open. For example, geometrical properties of the rank metric space are not well studied, and covering properties of rank metric codes have rece i ved littl e attention with the exception of o ur p revious work [1]. The ge ometrical properties, such as the intersection number , characterize funda mental properties of a metric space, and determine the pa cking and covering prope rties of codes defined in the metric space. Packing and covering properties are significant to the code design, d ecoding, and performance analysis of rank metric cod es. For instan ce, t he covering radius can b e vie we d as a measure o f performance: if the code is u sed for e rror correction, then the cov ering radius is the maximum weight of a correctable error vector . Th e covering radius of a code also gives a straightforward criterion for c ode optimality: if the covering radius of a code is at least equa l to it s minimum distan ce, then more codewords can be adde d wit hout altering the minimum distance, and hen ce the original code is not optimal. Althoug h lifti ngs of rank metric code s are nearly o ptimal c onstant-dimension codes, they have the largest possible covering radius [4], and hence they are not optimal con stant-dimension code s. This coveri ng property he nce is a crucial result for the design of error control codes for rand om line ar network coding. In this pa per , we focus on the ge ometrical properties of the rank metric s pace and covering properties of rank metric codes. W e first establish a n an alytical expression for the intersection of two balls wit h rank radii, and then deri ve an upp er bo und on the volume of the union of multiple balls with rank radii. Both results are novel to the best of our k nowledge. Using these geometrical properties, we de ri ve novel lower and upp er bounds on the minimum cardinality K R ( q m , n, ρ ) of a cod e in GF( q m ) n with rank covering radius ρ . The bo unds pre sented in this pa per are obtained bas ed on dif ferent approach es from those in [1], and they are the tightest bounds to the best of o ur knowl edge for many s ets of para meter v a lues. I I . P R E L I M I N A R I E S The rank weight of a vector x ∈ GF( q m ) n , denoted as rk( x ) , is de fined to be the maximum number of coordinates in x that are linearly independ ent over GF( q ) . The numbe r of vec tors of rank we ight r in GF( q m ) n is N r =  n r  α ( m, r ) , where α ( m, 0) def = 1 , α ( m, r ) def = Q r − 1 i =0 ( q m − q i ) , and  n r  def = α ( n, r ) /α ( r , r ) for r ≥ 1 [5]. The rank distance be tween x and y is defin ed as d R ( x , y ) = rk( x − y ) . If a vec tor x is at d istance at mo st ρ from a code C , we say C covers x wit h rad ius ρ . The rank c overing radius ρ of a Nov ember 4, 2018 DRAFT 3 code C is defi ned as max x ∈ GF( q m ) n d R ( x , C ) . Whe n n ≤ m , the minimum rank distance d R of a cod e o f length n and c ardinality M over GF( q m ) satisfies d R ≤ n − log q m M + 1 ; we refer to this bound as the Singleton bound for rank metric codes. T he equality is attained by a c lass of linea r ran k metric c odes called maximum rank distanc e (MRD) code s. I I I . G E O M E T R I C A L P RO P E RT I E S O F B A L L S W I T H R A N K R A D I I W e denote the intersection of two s pheres (balls, respectiv e ly) in GF( q m ) n with rank radii u and s and centers with distanc e w as J R ( u, s, w ) ( I ( u, s, w ) , respectiv ely). Lemma 1: W e ha ve J R ( u, s, w ) = 1 q mn N w n X i =0 N i K u ( i ) K s ( i ) K w ( i ) , (1) where K j ( i ) is a q -Krawtchouk polynomial [6]: K j ( i ) = j X l =0 ( − 1) j − l q lm + ( j − l 2 )  n − l n − j  n − i l  . (2) Although (1) is obtained by a direct application of [7, Chapter II, Theorem 3.6], we present it formally since it is a fundamental geometric property of the rank metric space . Since I ( u, s, w ) = u X i =0 s X j =0 J R ( i, j, w ) , (3) (1) a lso lea ds to an a nalytical expres sion for I ( u, s, w ) . In o rder to simplify n otations we denote I ( ρ, ρ, d ) as I ( ρ, d ) . Both (1) an d (3) a re ins trumental in ou r later de ri vations. W e den ote the volume of a ball with rank radius ρ as v ( ρ ) , and we now deriv e a bou nd on the volume of the u nion of any K balls with radius ρ , which will be instrumenta l in Sec tion IV. Lemma 2: The v olume of the union of any K balls with ran k radius ρ is at mo st B ( K ) = v ( ρ ) + l X a =1 ( q am − q ( a − 1) m )[ v ( ρ ) − I ( ρ, n − a + 1)] + ( K − q lm )[ v ( ρ ) − I ( ρ, n − l )] , (4) where l = ⌊ log q m K ⌋ . Pr oof: Let { v i } K − 1 i =0 denote the cen ters of K ba lls with rank radius ρ and let V j = { v i } j − 1 i =0 for 1 ≤ j ≤ K . The centers are labeled such that d R ( v j , V j ) is non-increas ing for j > 0 . For 1 ≤ a ≤ l and q m ( a − 1) ≤ j < q ma , we h ave d R ( v j , V j ) = d R ( V j +1 ) ≤ n − a + 1 by the Singleton bound. The center v j hence covers a t mos t v ( ρ ) − I ( ρ, n − a + 1) vectors that are not pre viously covered by V j . Nov ember 4, 2018 DRAFT 4 I V . B O U N D S O N C O V E R I N G C O D E S W I T H T H E R A N K M E T R I C W e cons ider cov ering codes in GF( q m ) n , and without los s of gene rality we assume n ≤ m due to transposition [1]. W e first de ri ve a lower bou nd on K R ( q m , n, ρ ) bas ed on the linear inequa lities satisfied by cov ering cod es. Pr opo sition 1: For 0 ≤ δ ≤ ρ , let T δ = m in P n i =0 A i ( δ ) , whe re the minimum is taken ov er all integer sequenc es { A i ( δ ) } which satisfy A i ( δ ) = 0 for 0 ≤ i ≤ δ − 1 , A δ ( δ ) ≥ 1 , 0 ≤ A i ( δ ) ≤ N i for δ + 1 ≤ i ≤ n , an d P n i =0 A i ( δ ) P ρ s =0 J R ( r , s, i ) ≥ N r for 0 ≤ r ≤ n . Then K R ( q m , n, ρ ) ≥ max 0 ≤ δ ≤ ρ T δ . Pr oof: Let C be a c ode with covering radius ρ . For any u ∈ GF( q m ) n at distance 0 ≤ δ ≤ ρ from C , let A i ( δ ) denote the number o f codew ords at d istance i from u . Then P n i =0 A i ( δ ) = | C | a nd we easily ob tain A i ( δ ) = 0 for 0 ≤ i ≤ δ − 1 , A δ ( δ ) ≥ 1 , and 0 ≤ A i ( δ ) ≤ N i for δ + 1 ≤ i ≤ n . Also, for 0 ≤ r ≤ n , all the vectors a t distanc e r from u a re covered, hence P n i =0 A i ( δ ) P ρ s =0 J R ( r , s, i ) ≥ N r . W e note that the no tation A i ( δ ) is used above since the constraints o n the sequence depend o n δ . Sinc e T δ is the solution of an integer linear programming, it is computationally infea sible to determine T δ for very large parameter values. W e now de ri ve an up per bound on K R ( q m , n, ρ ) by p roviding a nontri vial refinement of the greedy algorithm in [8]. Lemma 3: Let C be a code which covers at leas t q mn − u vectors in GF( q m ) n with radius ρ . Then for any k ≥ | C | , there exists a c ode with cardinality k whic h covers at least q mn − u k vectors, whe re u | C | = u and for k ≥ | C | u k +1 = u k −  u k v ( ρ ) min { q mn − k , B ( u k ) }  . (5) Thus K R ( q m , n, ρ ) ≤ min { k : u k = 0 } . Pr oof: The proof is by indu ction on k . By hyp othesis, C is a c ode with cardinality | C | which leav es u | C | vectors uncovered. S uppose the re exists a c ode with cardina lity k which leaves t k ≤ u k vectors uncovered, and de note the se t of uncovered vectors as T k . Let G be a graph where the vertex set is GF( q m ) n and two vertices are adjac ent if and only if the ir rank distanc e is at most ρ . Let A be the adjacen cy matrix of G and A k be the t k columns of A c orresponding to T k . The re are t k v ( ρ ) ones in A k , distrib u ted across | N ( T k ) | ro ws, where N ( T k ) is the neighborhoo d o f T k . By construction, N ( T k ) does not co ntain any codeword, hence | N ( T k ) | ≤ q mn − k . Also, by Lemma 2, | N ( T k ) | ≤ B ( t k ) ≤ B ( u k ) . Thus | N ( T k ) | ≤ m in { q mn − k , B ( u k ) } and there exists a row with at least l t k v ( ρ ) min { q mn − k ,B ( u k ) } m ones in A k . Adding the vector corresp onding to this row to the code, we obtain a code with cardinality k + 1 which lea ves at mo st t k − l t k v ( ρ ) min { q mn − k ,B ( u k ) } m ≤ u k +1 vectors u ncovered. Nov ember 4, 2018 DRAFT 5 The upper bound in L emma 3 is nontri vial s ince the s equen ce { u k } is monotonically decreas ing until it reaches 0 for k ≤ q mn . Since the sequence { u k } depend s on C on ly through the n umber of vectors c overed by C , to ob tain a tighter bound on K R ( q m , n, ρ ) ba sed on Lemma 3, it is nece ssary to find a code C which leaves a sma ller number of vectors uncovered. W e consider two choices for C below . Since any codew o rd can cover at most v ( ρ ) vectors uncovered b y the o ther codew ords, we have u ≥ q mn − v ( ρ ) | C | . In the first ch oice, we con sider the largest c ode C 0 that a chieves u = q mn − v ( ρ ) | C 0 | , which is a n ( n , n − a, a + 1) linear MRD code with a = m in { n, 2 ρ } . An a lternati ve choice would be to selec t a sup ercode of C 0 , that is, an MRD co de with a lar ger dimension. Since there may be intersection between balls of radius ρ around cod ew o rds, we now de ri ve a lower bou nd on the numbe r of vectors covered by only one codeword in an MRD code. Let us d enote the number of cod ew ords with rank r in an ( n, n − d + 1 , d ) linear MRD co de a s M ( d, r ) . Lemma 4: Let the vectors c j ∈ GF( q m ) n be sorted such that { c j } q m ( n − l +1) − 1 j =0 is an ( n, n − l + 1 , l ) MRD cod e for 1 ≤ l ≤ n . For all 1 ≤ k ≤ q mn − 1 , we de note { c j } k − 1 j =0 as C k and its minimum rank distance is gi ven by d k = n − ⌈ log q m ( k + 1) ⌉ + 1 . Then for d k ≥ 2 ρ + 1 , c k covers v ( ρ ) vectors n ot covered by C k . For d k ≤ 2 ρ , c k covers at least v ( ρ ) − P a l = d k µ l I ( ρ, l ) vectors no t covered by C k , where a = min { n, 2 ρ } , µ d k = min { M ( d k , d k ) , k } , and µ l = min n M ( d k , l ) , k − P l − 1 j = d k µ j o for l > d k . Pr oof: T he case d k ≥ 2 ρ + 1 is straightforward. W e assume d k ≤ 2 ρ h enceforth. W e denote the number of cod ew ords i n C k at distance l ( 0 ≤ l ≤ n ) fr om c k as M ( k ) l . Since a cod ew ord in C k at distance l from c k covers exactly I ( ρ, l ) vec tors also covered by c k , c k covers at leas t v ( ρ ) − P a l = d k M ( k ) l I ( ρ, l ) vectors that are not covered by C k . Since the value of M ( k ) l is un known, we gi ve a lower bo und o n the q uantity a bove. Since I ( ρ, l ) is a non-increa sing function of l [1], the s equenc e { µ l } a s defined above minimizes the value of v ( ρ ) − P a l = d k M ( k ) l I ( ρ, l ) under the c onstraints 0 ≤ M ( k ) l ≤ M ( d k , l ) and P n l = d k M ( k ) l = k . Thus , c k covers at least v ( ρ ) − P a l = d k µ l I ( ρ, l ) vectors not covered by C k . Pr opo sition 2: L et a = min { n, 2 ρ } , K 0 = q m ( n − a ) , and u K 0 = q mn − q m ( n − a ) v ( ρ ) . Consider two sequen ces { h k } and { u ′ k } , both of w hich are upper bounds on the number of vectors ye t to be covered by a c ode o f cardinality k , g i ven by h K 0 = u ′ K 0 = u K 0 and for k ≥ K 0 h k +1 = h k − v ( ρ ) + a X l = d µ l I ( ρ, l ) , u ′ k +1 = min  h k +1 , u ′ k −  u ′ k v ( ρ ) min { q mn − k , B ( u ′ k ) }  . Then, K R ( q m , n, ρ ) ≤ min { k : u ′ k = 0 } . Nov ember 4, 2018 DRAFT 6 m n ρ = 1 ρ = 2 ρ = 3 ρ = 4 ρ = 5 ρ = 6 2 2 3 1 3 2 4 1 3 11-16 4 1 4 2 7-8 1 3 40-64 4-7 1 4 293-722 10-48 3- 5 J 1 5 2 12-16 1 3 154-256 6-8 1 4 2267-409 6 33-256 4-8 1 5 34894- 2 17 233- 2773 I h 10 -32 3- 6 J 1 6 2 23-32 1 3 601-1024 11-16 1 4 17822- 2 15 124-256 6-16 1 5 550395 - 2 20 1770- 2 14 31-256 i 4 -16 1 6 173184 10- 2 26 27065- 40 1784 I 214- 4092 I 9- 154 J 3- 7 J 1 7 2 44-64 1 3 2372-409 6 20-32 1 4 141231 - 2 18 484-1024 i 10 -16 1 5 873528 9- 2 24 13835- 2 15 i 112 -1024 i 6 -16 1 6 549829 402- 2 30 42229- 2 22 i 1585 - 2 15 29- 708 I i 4 -16 1 7 349010 04402- 2 37 132054 50- 23354 9482 I i 23979 - 573 590 I 203- 5686 I h 9 - 211 J 3- 8 J T ABLE I B O U N D S O N K R (2 m , n, ρ ) , F O R 2 ≤ m ≤ 7 , 2 ≤ n ≤ m , A N D 1 ≤ ρ ≤ 6 . Pr oof: L et C 0 be an ( n, n − a, a + 1) MRD cod e over GF( q m ) . C 0 has c ardinality K 0 and covers q m ( n − a ) v ( ρ ) = q mn − u K 0 vectors in GF( q m ) n . By Lemma 4, adding k − K 0 codewords of an MRD code which properly contains C 0 leads to a code with cardinality k which covers at least q mn − h k vectors. On the other hand, u ′ k is the upper b ound on the n umber of vectors yet to be covered by a c ode of cardinality k that is obtained recursively either by applying the greedy approac h in the proof of Lemma 3 or by adding more cod ew ords base d on the MRD codes as de scribed in the proof o f L emma 4. The recursion o f u ′ k follo ws from this cons truction. Since this recursion is constructiv e , there exists a c ode with cardinality k wh ich lea ves at mos t u ′ k vectors u ncovered, a nd hence K R ( q m , n, ρ ) ≤ min { k : u ′ k = 0 } . Nov ember 4, 2018 DRAFT 7 W e now construct a new class o f covering codes with the rank metric. W e a ssume all the vectors in GF( q m ) n are expande d with respe ct to a given basis of GF( q m ) to m × n matrices ov e r GF( q ) . First, for any positiv e integer k , we denote S k def = { 0 , 1 , . . . , k − 1 } . For V ∈ GF( q ) m × n , I ⊆ S m , an d J ⊆ S n , we denote V ( I , J ) = ( v i,j ) i ∈ I ,j ∈ J , where the indexes are all so rted increasingly . Consider the code C which co nsists of all matrices C ∈ GF( q ) m × n with C ( S ρ , S n ) = 0 an d with at most n − ρ non zero columns. Pr opo sition 3: C ha s covering radius ρ and K R ( q m , n, ρ ) ≤ | C | = P n − ρ i =0  n i  ( q m − ρ − 1) i . Pr oof: First, the cardinality of C is g i ven by the number of vectors in GF( q m − ρ ) n with Hamming weight at mo st n − ρ . It remains to pro ve that C has rank c overing radius ρ . Sup pose V ∈ GF( q ) m × n has rk( V ( S ρ , S n )) = r ( 0 ≤ r ≤ ρ ), then the re exist I ⊆ S ρ and J ⊆ S n such that | I | = | J | = rk( V ( I , J )) = rk( V ( S m , J )) = rk( V ( I , S n )) = r . For ρ ≤ i ≤ m − 1 , let l i = V ( { i } , J ) V ( I , J ) − 1 ∈ GF( q ) 1 × r . Define U ( S ρ , S n ) = V ( S ρ , S n ) and U ( { i } , S n ) = l i V ( I , S n ) for all ρ ≤ i ≤ m − 1 . All the ro ws of U are in the row span of V ( I , S n ) , therefore rk( U ) = r . Also, U ( { i } , J ) = l i V ( I , J ) = V ( { i } , J ) for ρ ≤ i ≤ m − 1 and hence U ( S m , J ) = V ( S m , J ) . Thus there exists U ∈ GF( q ) m × n such that rk( U ) = r , U ( S ρ , S n ) = V ( S ρ , S n ) , and U ( S m , J ) = V ( S m , J ) . W e also construct U ′ ∈ GF( q ) m × n by setting U ′ ( S m , J ′ ) = V ( S m , J ′ ) and U ′ ( S m , S n \ J ′ ) = U ( S m , S n \ J ′ ) , where | J ′ | = ρ − r a nd J ∩ J ′ = ∅ . For C = V − U ′ , C ( S ρ , S n ) = 0 and C ( S m , J ∪ J ′ ) = 0 . Therefore, C ∈ C and d R ( V , C ) = rk( U ′ ) ≤ rk( U ) + | J ′ | = ρ . In order to illustrate the impro veme nt due to the bounds in this pap er , similar to [1], we provide the tightest lower a nd up per bounds on K R (2 m , n, ρ ) for 2 ≤ m ≤ 7 , 2 ≤ n ≤ m , and 1 ≤ ρ ≤ 6 in T able I. The improved entries in T a ble I due to the results in this paper are b oldface, and a re ass ociated with letters ind icating the sources. The lower bound o n K R ( q m , n, ρ ) in [1, Proposition 8] is based on I ( ρ, d ) . Howe ver , due to the lack of analytical expres sion for I ( ρ, d ) , in [1] we were able to comp ute I ( ρ, d ) only for small values, using exhausti ve search. Using (1)-(3), we ca lculate I ( u, s, w ) and hence the bound in [1, Propos ition 8] for a ny set of parame ter values, and the improved entries for this reas on are a ssociated with the lower case letter i. T he lo we r ca se letter h and the upper case letters I a nd J correspond to improvements du e to Propos itions 1, 2, and 3, resp ectiv ely . The unmarked entries in T ab le I are the same as those in [1]. R E F E R E N C E S [1] M. Gadouleau and Z . Y an, “Packing and covering properties of rank metric codes, ” IEEE Tr ans. Info. Theory , vol. 54, no. 9, pp. 387 3–3883 , September 2008. 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