Optimal codes for correcting a single (wrap-around) burst of errors

Optimal co des for correcti ng a single (wrap-around) burst of eras ures Henk D.L. Hollma nn and Ludo M.G. M. T olhuizen ∗ September 4, 2021 Abstract In 2007, Martinian and T r ott pr esen ted co des for correcting a bu rst of erasures with a minim um deco ding dela y . Their construction emp lo ys [ n, k ] co des that can correct a ny burst of erasures (including wrap -aroun d bu rsts) of length n − k . They raised the question if suc h [ n , k ] co des exist for all in tegers k and n with 1 ≤ k ≤ n and all fields (in particular, for the binary field). In this note, w e answ er this question affirmativ ely by giving t w o recurs ive constructions and a dir ect one. ∗ The authors are with Philips Research Lab ora tories, Prof. Holstlaa n 4, 5656 AA Eindhov en, The Netherlands; e-mail: { henk.d.l.ho llmann,ludo.tolhuizen } @philips.com 1 1 In tro d uction In [1], Martinian and T rott presen t co des for correcting a burst of erasures with a minim um deco ding dela y . Their construction emplo ys [ n, k ] co des that can correct an y burst o f erasures (including wrap-around bursts) of length n − k . Examples of suc h co des are MDS co des and cyclic co des. The question is raised in [1] if suc h [ n, k ] co des exist for all in tegers k and n with 1 ≤ k ≤ n a nd all fields (in particular, ov er the binary field). In this note, we answ er this question affirmative ly b y giving tw o recursiv e constructions and a direct one. Throughout this note, all matrices and co des are ov er the (fix ed but arbitrary) finite field F , and we restrict ourselv es to linear co des. Ob viously , a co de of length n can correct a pattern E of erasures if and only if an y co de- w or d can b e uniquely recov ered fro m its v alues in the ( n − | E | ) p o sitions outside E . As a consequenc e, if an [ n, k ] co de can correct a pattern E of erasures, then n − | E | ≥ k , i.e. , | E | ≤ n − k . W e call an [ n, k ] co de optimal if it can correct an y burst of erasures (including wrap-around bursts) of length n − k . 1 Equiv a lently , an [ n, k ] co de is optimal if knowle dge of any k (c yclically) consecutiv e sym b ols from a co dew ord allows one to uniquely reco v er that co dew ord, o r , in co ding parlance, if eac h of the n sets of k (cyclically) consecutiv e co dew ord p ositions forms an informatio n set. W e call a k × n matrix go o d if any k cycli- cally c onsecutiv e columns of G are indep enden t. It is easy to see that a co de is optimal if and only if it has a go o d generator matrix. Throughout this note, we denote with I k the k × k identit y matrix, and with X T the transp ose of the matrix X . 2 A recursi v e constr u ction of opti mal co des In this se ction, w e give a recursiv e construction of go o d matrices , and henc e of optimal co des. W e start with a simple dualit y result. Lemma 2.1 L et C b e an [ n, k ] c o de, and let C ⊥ b e its dual. If I ⊂ { 1 , . . . , n } has size k and is an information set for C , then I ∗ = { 1 , . . . , n } \ I is an information set for C ⊥ . Pro of: By contradiction. Suppose that I ∗ is not an information set for C ⊥ . Th en there is a non-zero w ord x in C ⊥ that is zero in the p ositions indexed by I ∗ . As x is in C ⊥ , for an y word c ∈ C we hav e that 0 = n X i =1 x i c i = X i ∈ I x i c i . 1 A mor e precise terminolog y would b e ” optimal for the correction of a sing le (wrap-ar ound) burst of erasures ”, but we o pted for just ”o ptimal” for notational c onv e nienc e . 1 As a conseq uence, there are k -tuples that do not occur in I in an y w ord of C , a con t ra- diction. W e conclude tha t I ∗ is a n infor ma t ion set fo r C ⊥ . ✷ As a consequence, w e ha v e the follo wing. Corollary 2.2 A line ar c o de is optimal if and only if its dual i s optimal. Our first theorem sho ws ho w t o construct a go o d k × ( k + n ) mat rix fro m a go o d k × n matrix. Theorem 2.3 L et G = ( I k P ) b e a go o d k × n matrix. T hen G ′ = ( I k I k P ) is a go o d k × ( k + n ) ma trix. Pro of: An y k cyclically consecutiv e columns in G ′ either are k differen t unit v ectors, or k cyc lically consecutiv e columns of G . ✷ Our next theorem sho ws how to construct a go o d n × (2 n − k ) mat rix from a go o d k × n matrix. Theorem 2.4 L et G = ( I k P ) b e a go o d k × n ma trix. T he the fol low ing n × (2 n − k ) matrix G ′ is go o d G ′ =  I n − k 0 I n − k 0 I k P  . Pro of: As G is go o d, Corollary 2.2 implies that the g enerator matrix ( − P T I n − k ) of the dual of the co de generated b y G is go o d. By cyclically shifting the columns of this matrix o v er ( n − k ) p ositions to the righ t, w e obtain the go o d matrix ( I n − k − P T ). Theorem 1 implies that ( I n − k I n − k − P T ) is go o d, and so the matrix H =  I n − k − P T I n − k  obtained by cyclically shifting the columns of the former matrix ov er n p ositions, is go o d. Clearly , after m ultiplying the columns of a g o o d mat r ix with non- zero field elemen ts, w e obtain a go o d matrix; as a conse quence, H ′ = ( − I n − k − P T I n − k ) is go o d. As H ′ is a g o o d full- rank parity c heck matrix of the co de generated b y G ′ , this latter matrix is go o d. ✷ Remark The construction from Theorem 2.4 also o ccurs in the pro of o f [1, Thm.1]. The cons truction from Theorem 2.3 incre ases the co de length a nd fixes its dimension; the construction from Theorem 2.4 also increases the co de length, but fixes its redun- dancy . These constructions can b e combine d to give a recu rsiv e construction of optimal [ n, k ] co de for all k and n . The follow ing definition is instrumental in making this explicit. Definition 2.5 F or p ositive inte gers r and k , we r e cursively define the k × r matrix P k ,r as f o l lows: P k ,r =         I r P k − r,r  if 1 ≤ r < k , I k if r = k , ( I k P k ,r − k ) if r > k . 2 Theorem 2.6 F or e ach p os i tive inte g e r k , the matrix I k is go o d . F or al l inte gers k and n with 1 ≤ k < n , the k × n matrix ( I k P k ,n − k ) is go o d. Pro of: The first statemen t is ob vious. The second statemen t will b e pro v ed by ind uction on k + n . It is easily v erified that it is true for k + n = 3 . No w assume that the statemen t is true for all integers a, b with 1 ≤ a ≤ b and a + b < k + n . W e consider three cases. If 2 k < n , then b y induction h yp othesis ( I k P k ,n − 2 k ) is go o d. By Theorem 2.3, ( I k I k P k ,n − 2 k ) = ( I k P k ,n − k ) is also go o d. If 2 k = n , t hen ( I k P n − k ) = ( I k P k ,k ) = ( I k I k ), whic h ob viously is a go o d matrix. If k < n and 2 k > n , the induction hy p othesis implies that ( I 2 k − n P 2 k − n ,n − k ) is a go o d (2 k − n ) × k matrix. By Theorem 2 .4,  I n − k 0 I n − k 0 I 2 k − n P 2 k − n ,n − k  = ( I k P k ,n − k ) is a lso go o d. ✷ Example 2.7 Theorem 2.6 implies that ( I 28 P 28 , 17 ) is a go o d 28 × 45 matrix. According t o t he definition, P 28 , 17 =  I 17 P 11 , 17  . Again according to the definition, P 11 , 17 = ( I 11 P 11 , 6 ). Con t inuing in this fashion, P 11 , 6 =  I 6 P 5 , 6  . Finally , P 5 , 6 = ( I 5 P 5 , 1 ), and, as c an b e readily seen by induction on k , P k , 1 is the all- one v ector of heigh t k . Putting this alto gether, w e find that the follo wing 28 × 45 matrix G is g o o d: G =       I 6 0 0 0 0 I 6 0 0 0 I 5 0 0 0 0 I 5 0 0 0 I 6 0 0 0 0 I 6 0 0 0 I 6 0 I 6 0 I 6 0 0 0 0 I 5 0 I 5 P 5 , 6       , where P 5 , 6 = ( I 5 1 ), where 1 denotes the all- one column vec tor. T o close this section, we remark tha t with an induction argumen t it can be sho wn that for a ll p ositiv e integers k and r , w e ha v e P k ,r = P T r,k . 3 Adding o ne c olumn to a go o d matrix In Theorem 2.3, w e added k columns to a go o d k × n matrix to obtain a go o d k × ( k + n ) matrix. In this section, w e will sho w that it is alw a ys p ossible to add a single column to 3 a go o d k × n matrix in suc h a w ay that the resulting k × ( n + 1) matrix is g o o d; w e a lso sho w that the in the binary case, there is a unique column that can b e added. The desired result is a direct consequence of the following observ ation, which ma y be of indep enden t in terest. Lemma 3.1 L et F b e any fie ld, and let a 1 , a 2 , . . . , a 2 k − 2 b e a se quenc e of ve ctors in F k such that a i , a i +1 , . . . , a i + k − 1 ar e indep endent over F for i = 1 , . . . , k − 1 . F or i = 1 , . . . , k , let b i b e a nonzer o ve ctor ortho gonal to a i , a i +1 , . . . , a i + k − 2 . Then b 1 , . . . , b k ar e indep enden t over F . Pro of: F or i = 1 , . . . , k , w e define V i := span { a i , . . . , a i + k − 2 } . F or an inte rv al [ i + 1 , i + s ] := { i + 1 , i + 2 , . . . , i + s } , with 0 ≤ i < i + s ≤ k , w e let V [ i +1 ,i + s ] = V i +1 ∩ · · · ∩ V i + s denote the inters ection o f V i +1 , . . . , V i + s . Note that by definition V [ i,i ] = V i = b ⊥ i . W e claim that V [ i +1 ,i + s ] = span { a i + s , . . . , a i + k − 1 } . This is easily prov en b y induction o n s : ob viously , the claim is true fo r s = 1; if it holds for a ll s ′ ≤ s , then V [ i +1 ,i + s +1] = V [ i +1 ,i + s ] ∩ V i + s +1 = span { a i + s , . . . , a i + k − 1 } ∩ span { a i + s +1 , . . . , a i + s + k − 1 } , hence V [ i +1 ,i + s ] certainly con tains a i + s +1 , . . . , a i + k − 1 and do es not con t a in a i + s , since by assumption a i + s / ∈ span { a i + s +1 , . . . , a i + s + k − 1 } . So b y our claim it follo ws that { 0 } = V [1 ,k ] = V 1 ∩ · · · V k = b ⊥ 1 ∩ · · · b ⊥ k , hence b 1 , . . . , b k are indep enden t. ✷ As an immediate consequence, we ha v e the f ollo wing. Theorem 3.2 L et M b e a go o d k × n matrix over GF( q ) . Ther e ar e p r e cisely ( q − 1) k ve ctors x ∈ GF( q ) k such that the matrix ( M x ) is go o d. 4 Pro of: Let M = ( m 0 , m 1 , . . . , m n − 1 ) hav e columns m 0 , . . . m n − 1 ∈ GF ( q ) k . W e wan t to find a ll v ectors x ∈ GF( q ) k with the prop ert y that the k v ectors m n − i , . . . , m n − 1 , x, m 0 , . . . , m k − i − 2 (1) are indep enden t, for all i = k − 1 , k − 2 , . . . , 0. So, for i = k − 1 , k − 2 , . . . , 0, let b i b e a nonzero v ector orthogonal to m n − i , . . . , m n − 1 , m 0 , . . . , m k − i − 2 ; since M is go o d, the k − 1 v ectors m n − i , . . . , m n − 1 , m 0 , . . . , m k − i − 2 are independen t, a nd hence the ve ctors in (1) are indep enden t if and only if ( x, b i ) = λ i 6 = 0. Again since M is go o d, the 2 k − 2 v ectors m n − k +1 , . . . , m n − 1 , m 0 , . . . , m k − 2 satisfy the conditions in Lemma 3.1, hence the v ectors b 0 , . . . , b k − 1 are independen t. So for eac h c hoice of λ = ( λ 0 , . . . , λ k − 1 ) with λ i 6 = 0 for eac h i , there is a unique v ector x f o r whic h ( x, b i ) = λ i , and these v ectors x are precisely the ones for w hic h ( M x ) is go o d. ✷ 4 Explicit const ruction of go o d matrices By starting with the k × k iden t ity matrix, and repeatedly applying Theorem 3.2, w e find that for each field F and all p ositiv e in tegers k and n with n ≥ k , there exists a k × n matrix G such that (1) the k leftmost columns of G form the k × k iden tity matrix, and (2) fo r each j , k ≤ j ≤ n , the j leftmost columns o f G form a go o d k × j matrix. Note that T heorem 3.2 implie s that for the binar y field, these ma t rices are unique. It turned out that they ha v e a simple recursiv e structure, whic h inspired our general con- struction. In this section, w e giv e, f o r all p ositiv e in t egers k and n with k ≤ n , an explicit construction of k × n matrices ov er Z p , the field of integers mo dulo p , that satisfy t he ab ov e prop erties ( 1 ) and (2) . Note that such matrices also satisfy (1 ) and (2) f or extension fields of Z p . W e start with describing the result for p = 2. Let M 1 b e the matrix M 1 =  1 0 1 1  , (2) and for m ≥ 1, let M m +1 b e the giv en as M m +1 =  M m 0 M m M m  . (3) Clearly , M m is a binary 2 m × 2 m matrix. The relev ance of the matrix M m to our problem is explained in the follow ing theorem. 5 Theorem 4.1 L et k and r b e two p o sitive i n te gers, and let m b e the sm al lest inte ger such that 2 m ≥ k and 2 m ≥ r . L et Q b e the k × r matrix r esiding in the lower left c orner of M m . Then for e ach inte g e r j for which k ≤ j ≤ k + r , the j leftmost c olumns of the matrix ( I k Q ) form a go o d binary k × j matrix. Theorem 4.1 is a consequence from our results f o r the general case in t he remainder o f this section. W e no w define the matrices that ar e relev an t for constructing go o d matrices o v er Z p . Definition 4.2 L et p b e a prime numb er, an d let k , r b e p ositive inte gers. L et m b e the smal lest inte ger such that p m ≥ r and p m ≥ k . The k × r ma trix Q k ,r is d e fine d as Q k ,r ( i, j ) =  p m − k + i − 1 j − 1  for 1 ≤ i ≤ k , 1 ≤ j ≤ r . In T heorem 4.8 w e will show that the matrix ( I k Q k ,r ) is go o d ov er Z p . But first, w e de riv e a recursiv e property of the Q - matrices. T o this aim, we need some w ell-kno wn results on binomial co efficien ts mo dulo p . Lemma 4.3 L et p b e a pri m e numb er, and le t m b e a p ositive inte ger. F or any inte ger i with 1 ≤ i ≤ p m − 1 , we have that  p m i  ≡ 0 mo d p . Pro of: The following pro o f w as pointed out to us by our colleague R onald Rietman. Let 1 ≤ i ≤ p m − 1 . W e hav e that  p m i  = p m  p m − 1 i − 1  i . In the ab ov e represen ta tion of  p m i  , the nominator con ta ins at least m fa ctors p , while the denominato r contains a t most m − 1 factors p . ✷ Lemma 4.4 L et p b e a p ri m e numb er, and let m b e a p osi tive inte ge r. Mor e ov e r, let i, j, k , ℓ b e inte ge rs such that 0 ≤ i, k ≤ p − 1 and 0 ≤ j, ℓ ≤ p m − 1 . Then we have that  ip m + j k p m + ℓ  ≡  i k  j ℓ  mo d p. Pro of: This is a direct consequence o f Lucas’ theorem (see for example [2 , Thm. 13 .3 .3]). W e give a short direc t proof. Clearly ,  ip m + j k p m + ℓ  is the coefficien t of z k p m + ℓ in (1 + z ) ip m + j . No w we note that (1 + z ) ip m + j = (1 + z ) ip m (1 + z ) j =  (1 + z ) p m  i (1 + z ) j . It follo ws fro m Lemma 4.3 that (1 + z ) p m ≡ 1 + z p m mo d p , and so (1 + z ) ip m + j ≡ (1 + z p m ) i (1 + z ) j mo d p. Hence, mo dulo p , t he co efficien t of z k p m + ℓ in (1 + z ) ip m + j equals  i k  j ℓ  . ✷ 6 Corollary 4.5 L et p b e a prime, and let m b e a p o sitive inte ger. L e t a, b, c, d b e i nte gers such that 0 ≤ a, c ≤ p − 1 and 1 ≤ b, d ≤ p m . The n we have Q p m +1 ,p m +1 ( ap m + b, cp m + d ) ≡  a c  Q p m ,p m ( b, d ) mo d p. Pro of: According to the definition o f Q p m +1 ,p m +1 , w e hav e that Q p m +1 ,p m +1 ( ap m + b, cp m + d ) =  ap m + b − 1 cp m + d − 1  , and Q p m ,p m ( b, d ) =  b − 1 d − 1  . The corollary is now obtained b y application of Lemma 4.4. ✷ In words, Theorem 4.5 states that Q p m +1 ,p m +1 can b e considered as a p × p blo ck matrix, for which eac h blo c k is a m ultiple of Q p m ,p m . F or example, for p = 3 , we obtain Q 3 m +1 , 3 m +1 =    0 0   0 1   0 2   1 0   1 1   1 2   2 0   2 1   2 2    × Q 3 m , 3 m =   Q 3 m , 3 m 0 0 Q 3 m , 3 m Q 3 m , 3 m 0 Q 3 m , 3 m 2 Q 3 m , 3 m Q 3 m , 3 m   . F or p = 2, w e obtain the relatio n in (3). T aking a = p − 1 and c = 0 in Theorem 4.5, w e se e that ov er Z p , the p m × p m blo c k in the lo w er left hand corner of Q p m +1 ,p m +1 equals Q p m ,p m . Definition 4.2 implies Q k ,r is the k × r matrix residing in the lo we r left hand corner of Q p m ,p m , where m is the smallest in teger that suc h that p m ≥ k and p m ≥ r . The ab ov e observ ations imply that whenev er k ′ ≥ k and r ′ ≥ r , then ov er Z p , the matrix Q k ,r is the k × r submatrix in the low er left hand corner of Q k ′ ,r ′ . In particular, Q k ,r +1 can b e obtained b y a dding a column to Q k ,r . W e now state and prov e results on the in vertibilit y in Z p of certain submatrices o f Q k ,r , that will b e used to pro v e our main result in Theorem 4.8. Lemma 4.6 L et n ≥ 0 and b ≥ 1 . The b × b matrix V b with V b ( i, j ) =  n + i − 1 j − 1  for 1 ≤ i, j ≤ b has an inte ger inverse. Pro of: By induction on b . F or b = 1, this is ob vious. Next, let b ≥ 2. Let S b e t he b × b mat r ix with S ( i, j ) =    1 if i = j, − 1 if i ≥ 2 and i = j + 1 , 0 otherwise. The matrix S has an in teger inv erse: it is easy to c hec k t hat S − 1 ( i, j ) = 1 if i ≥ j , and 0 otherwise. W e ha ve that ( S V b )(1 , j ) = V b (1 , j ) =  n j − 1  , a nd 7 ( S V b )( i, j ) = V b ( i, j ) − V b ( i − 1 , j ) =  n + i − 1 j − 1  −  n + i − 2 j − 1  =  n + i − 2 j − 2  for 2 ≤ j ≤ b. In o ther w ords, S V b is o f the form S V b =  1 A 0 V b − 1  . By induction h yp othesis, V b − 1 has an in teger inv erse, and so V b S has an in teger inv erse (namely the matrix  1 − AV − 1 b − 1 0 V − 1 b − 1  ). As S has an in teger inv erse, w e conclude that V b has a n integer inv erse. ✷ Lemma 4.7 L et p b e a prime numb er, and let a ≥ 0 and b ≥ 1 b e inte gers such that a + b ≤ p m . The b × b matrix W b with W b ( i, j ) =  p m − 1+ i − b a + j − 1  for 1 ≤ i, j ≤ b is inv e rtible over Z p . Pro of: Similarly to the pro of of Lemma 4.6 , we apply induction on b . F or b = 1, the w e ha ve the 1x1 matrix with en try  p m − 1 a  . By induction on i , using that  p m − 1 i  =  p m i  −  p m − 1 i − 1  and employin g Lemma 4.3, w e readily find t hat  p m − 1 i  ≡ ( − 1) i mo d p for 0 ≤ i ≤ p m − 1. As a consequenc e, the lemma is tr ue for b = 1. No w let b ≥ 2. W e define the b × b matrix T b y T ( i, j ) =    1 if i = j 1 if j ≥ 2 and i = j − 1 0 otherwise It is easy to chec k T has an integer inv erse, and that T − 1 ( i, j ) = ( − 1) i − j if i ≤ j and 0 otherwise. In order to sho w t ha t W b is in v ertible in Z p , it is thus sufficien t to sho w t hat W b T is in v ertible in Z p . By direct computation, w e ha v e that ( W b T )( i, 1) = W b ( i, 1), and ( W b T )( i, j ) = W b ( i, j )+ W b ( i, j − 1) =  p m − 1 + i − b a + j − 1  +  p m − 1 + i − b a + j − 2  =  p m + i − b a + j − 1  . In particular, ( W b T )( b, 1) =  p m − 1 a  ≡ ( − 1) a mo d p , and f or 2 ≤ j ≤ b , w e ha v e that ( W b T )( b, j ) =  p m a + j − 1  ≡ 0 mo d p . W e th us ha v e t ha t W b T ≡  A W b − 1 ( − 1) a 0  mo d p. As W b − 1 is inv ertible ov er Z p , the matr ix W b T (and hence the matrix W b ) is inv ertible o v er Z p . ✷ Remark The matrix in Lemma 4.7 need not hav e an in t eger in v erse. F or example, tak e p = 2 , m = 2 , a = 1 and b = 2. The matrix W 2 equals   2 1   3 1   2 2   3 2   =  2 3 1 3  , 8 and so W − 1 2 =  1 − 1 − 1 3 2 3  . No t e that mo dulo 2, W 2 equals  0 1 1 1  , confirming that W 2 do es ha v e an in vers e in the in tegers mo dulo p = 2. W e a r e now in a p osition to prov e the main result of this section. Theorem 4.8 L et k and r b e p ositive inte g e rs. F or j = k, k + 1 , . . . k + r , the matrix c onsisting o f the j le f tmo s t c olumn s of the matrix ( I k Q k ,r ) is go o d over Z p . Pro of: W e denote t he matrix ( I k Q k ,r ) b y G , and the i -th column of G b y g i . Let k ≤ j ≤ k + r . T o sho w that the matrix consisting of the columns 1,2,. . . , j of G is go o d, w e sho w that for 1 ≤ i ≤ j , the v ectors g i , g i +1 , . . . , g i + k − 1 are indep enden t o v er Z p , where the indices are counted mo dulo j . This is ob vious if j = k and if i = 1, so w e assume t ha t j ≥ k + 1 and i ≥ 2. W e distinguish b et w een t w o cases. (1) 2 ≤ i ≤ k . The v ectors to consider are e i , . . . , e k , g k +1 , . . . , g i + k − 1 (if i + k − 1 ≤ j ), or e i , . . . , e k , g k +1 , . . . , g j , e 1 , . . . , e k − j + i − 1 (if i + k − 1 ≥ j + 1). W e define b :=min( i − 1 , j − k ). The v ectors under consideration are indep enden t if the b × b matr ix consisting of the b leftmost columns of Q k ,r , restricted to rows i − b, i − b + 1 , . . . , i = 1, is in v ertible in Z p . This follo ws f rom Lemma 4 .6. ( 2) i ≥ k + 1. The v ectors to consider a re g i , . . . , g i + k − 1 (if i + k − 1 ≤ j ), or g i , . . . , g j , e 1 , . . . , e k − j + i − 1 (if i + k − 1 ≥ j + 1 ) . W e define b :=min( k, j − i + 1). The v ectors under c onsideration are independen t if the b × b matrix consisting of the b bottom en tries of the columns i − k + 1 , i − k + 2 , . . . , i − k + b of Q k ,r is in v ertible in Z p . This follows from Lemma 4.7. ✷ References [1] E. Martinian and M. T rott, ” Dela y-Optimal Burst Erasure Co de Construction”, ISIT 2007, Nice, F ra nce, June 24-29 , 2007, pp. 1006–101 0 . [2] R.E. Blahut, The ory and Pr a c tic e of Err or C ontr ol Co d es , Addison W esley , 1983 . 9