On a one-parameter family of Riordan arrays and the weight distribution of MDS codes

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1 On a one-parameter famly of Roran arrays an the weght strbuton of MDS coes Paul Barry School of Scence Waterfor Insttute of Technology Irelan pbarry@wte Patrck Ftzpatrck Department of Mathematcs Unversty College Cork Irelan ftzpat@ucce Abstract We use the formalsm of the Roran group to stuy a one-parameter famly of lower-trangular matrces relate to the weght strbuton of maxmum stance separable coes We obtan factorzaton results for these matrces We then erve alternatve expressons for the weght strbuton of MDS coes We efne relate weght ratos an show that they satsfy a certan lnear recurrence Introucton In ths note, we report on a one-parameter famly of transformaton matrces whch can be relate to the weght strbuton of maxmum stance separable (MDS coes Regare as transformatons on nteger sequences, they are easy to escrbe both by formula (n relaton to the general term of a sequence an n terms of ther acton on the ornary generatng functon of a sequence To acheve ths, we use the language of the Roran group of nfnte lower-trangular nteger matrces They are also lnke to several other known transformatons, most notably the bnomal transformaton 2 Error-correctng coes Maxmum stance separable coes are a specal case of error-correctng coe By errorcorrectng coe, we shall mean a lnear coe over F q GF (q, that s, a vector subspace C of Fq n for some n > 0 If C s a k-mensonal vector subspace of Fq n, then the coe s

2 escrbe as a q-ary [n, k]-coe The elements of C are calle the coewors of the coe The weght w(c of a coewor c s the number of non-zero elements n the vector representaton of c An [n, k] coe wth mnmum weght s calle an [n, k, ] coe A coe s calle a maxmum separable coe f the mnmum weght of a non-zero coewor n the coe s n k +, that s, n k + The Ree-Solomon famly of lnear coes s a well-known famly of MDS coes An mportant characterstc of a coe s ts weght strbuton Ths s efne to be the set of coeffcents A 0, A,, A n where A s the number of coewors of weght n C The weght strbuton of a coe plays a sgnfcant role n calculatng probabltes of error Except for trval or small coes, the etermnaton of the weght strbuton s normally not easy The MacWllams entty for general lnear coes s often use to smplfy ths task The specal case of MDS coes proves to be tractable Usng the MacWllams entty [2] or otherwse [3], [9], we obtan the followng equvalent results Proposton The number of coewors of weght, where n k + n, n a q-ary [n, k] MDS coe s gven by where mn n k + A ( n mn ( (q ( q mn ( 0 ( n ( ( (q mn+ (2 0 ( n mn ( We note that the last expresson can be wrtten as 0 ( (q mn+ (3 ( mn n ( A ( mn (q + (4 + mn by a smple change of varable We have A 0, an A 0 for n k The term ( n s a scalng term, whch also ensures that A 0 for > n In the sequel, we shall stuy a one-parameter famly of Roran arrays assocate to the equvalent summaton expressons above 3 Transformatons on nteger sequences an the Roran Group We shall ntrouce transformatons that operate on nteger sequences An example of such a transformaton that s wely use n the stuy of such sequences s the so-calle Bnomal 2

3 transform [0], whch assocates to the sequence wth general term a n the sequence wth general term b n where n ( n b n a (5 0 If we conser the sequence wth general term a n to be the column vector a (a 0, a, then we obtan the bnomal transform of the sequence by multplyng ths (nfnte vector by the lower-trangle matrx B whose (, -th element s equal to ( : B Ths transformaton s nvertble, wth a n n 0 ( n ( n b (6 We note that B correspons to Pascal s trangle Its row sums are 2 n, whle ts agonal sums are the Fbonacc numbers F (n + If B k enotes the k th power of B, then the n th term of B k a where a {a } s gven by n 0 kn ( n a If A(x s the ornary generatng functon of the sequence a n, then the generatng functon of the transforme sequence b n s A( x The bnomal transform s an element of x x the Roran group, whch can be efne as follows The Roran group [, 4, 8], s a set of nfnte lower-trangular nteger matrces, where each matrx s efne by a par of generatng functons g(x + g x + g 2 x 2 + an f(x f x + f 2 x 2 + where f 0 [8] We sometmes wrte f(x xh(x where h(0 0 The assocate matrx s the matrx whose -th column s generate by g(xf(x (the frst column beng nexe by 0 The matrx corresponng to the par f, g s enote by (g, f or R(g, f The group law s then gven by (g, f (h, l (g(h f, l f (7 The entty for ths law s I (, x an the nverse of (g, f s (g, f (/(g f, f where f s the compostonal nverse of f To each Roran array as efne above s assocate an nteger sequence A {a } wth a 0 0 such that every element n+,k+ of the array (not lyng n column 0 or row 0 can be expresse as a lnear combnaton wth coeffcents n A of the elements n the preceng row, startng from the preceng column: n+,k+ a 0 n,k + a n,k+ + a 2 n,k+2 + 3

4 A {a } s calle the A-sequence of the array, an may be calculate accorng to A(x [h(t t xh(t ] A Roran array of the form (g(x, x, where g(x s the generatng functon of the sequence a n, s calle the sequence array of the sequence a n Its general term s a n k If M s the matrx (g, f, an a (a 0, a, s an nteger sequence wth ornary generatng functon A (x, then the sequence Ma has ornary generatng functon g(xa(f(x Example 2 The bnomal matrx B s the element (, x of the Roran group It x x has general element ( n k More generally, B m s the element (, x of the Roran mx mx group, wth general term ( n k m n k It s easy to show that the nverse B m of B m s gven by (, x +mx +mx The row sums of the matrx (g, f have generatng functon g(x/( f(x whle the agonal sums of (g, f have generatng functon g(x/( xf(x Many nterestng examples of Roran arrays can be foun n Nel Sloane s On-Lne Encyclopea of Integer Sequences, [5], [6] Sequences are frequently referre to by ther OEIS number For nstance, the matrx B s A Introucng the one-parameter famly of MDS transforms In ths secton, we shall frequently use n an k to aress elements of nfnte arrays Thus the n, k-th element of an nfnte array T refers to the element n the n-th row an the k-th column Row an column nces wll start at 0 Ths customary use of n, k, shoul not cause any confuson wth the use of n, k above to escrbe [n, k] coes We efne T m to be the transformaton represente by the matrx ( + x T m mx, x + x where m N For nstance, we have T ( + x x, x + x Ths trangle s A330, whch has row sums, 3, 4, 4, 4, A33 wth generatng functon (+x2 In general, the row sums of T x m have generatng functon (+x2 Note also mx 4

5 that ( x T 0 + x, x wth general term ( n 2 n k ( n k We can calculate the A-sequence for T m as follows [ ] A(x t xh(t + t [ ] t x( + t + t [ + t t x ] x + x x x Thus every element n T m s gven by the fference of the two elements above t, e: T m (n +, k + T m (n, k T m (n, k + Proposton 3 For each m, T m s nvertble wth ( T x m (m + x, x Proof Let T m (g, f Ths exsts snce T m s an element of the Roran group Then ( (g, f + x mx, x (, x + x Hence an f + f x f x x g g f g m f (m + x + f Corollary 4 The general term of T m T m (n, k ( n n k s gven by ( n 2 (m + n k 5

6 Proof We have ( k x T m (n, k [x n ]( (m + x x [x n k ] ( x (m + k [xn k ] ( x ( ( k k k ( n k (m + n k n k ( ( n n 2 (m + n k n k ( n k Our man goal n ths secton s to fn expressons for the general term T m (n, k of T m To ths en, we exhbt certan useful factorzatons of T m Proposton 5 We have the followng factorzatons of the Roran array T m : ( + x T m mx, x + x ( ( + x, x mx, x + x ( ( x, + x (m + x, x ( ( mx, x x + x, + x ( ( + x, x + x x (m + x, x Proof Each of the assertons s a smple consequence of the prouct rule (equaton (7 for Roran arrays For nstance, ( ( ( x, + x (m + x, x x (m + x, + x +x ( + x + x (m + x, x + x ( + x mx, x T m + x The other assertons follow n a smlar manner The last asserton, whch can be wrtten ( T m B x (m + x, x, 6

7 s a consequence of the fact that the prouct BT m takes on a smple form We have ( ( BT m x, x + x x mx, x + x ( x + x x x x m x, + x x x ( x (m + x, x We can nterpret ths as the sequence array for the partal sums of the sequence (m + n, that s, the sequence array of (m+n+ Thus T (m+ m s obtane by applyng B to ths sequence array We note that the nverse matrx (BT m takes the specal form (( x( (m + x, x ( (m + 2x + (m + x 2, x Thus ths matrx s tr-agonal, of the form (m m + (m (BT m 0 m + (m m + (m m + (m + 2 Corollary 6 The general term of the array T m s n ( n T m (n, k ( n ((m + k+ /m, m 0 k Proof By the last proposton, we have ( T m B x The general term of B (, ( x +x +x s ( n k n s (m+n k+ (m+ Corollary 7 T m (n, k (m + x, x k whle that of the secon Roran array The result follows from the prouct formula for matrces n ( n m ( n k+ n k ( n m ( n k + m + k m k Equvalently, ( n (m k T m (n, k ( n n k k 0 0 ( n ( n k (m + + k 7

8 Ths last result makes event the lnk between the Roran array T m an the weght strbuton of MDS coes Thus, base on equaton (3, we have T q (, mn mn ( ( q mn + q Hence ( n (q T q (, mn ( ( n ( (q mn+ mn A an thus ( n A (q T q (, mn (8 We now fn a number of alternatve expressons for the general term of T m whch wll gve us a choce of expressons for the weght strbuton of an MDS coe Proposton 8 T m (n, k n k ( + k 2 ( m n k 0 n k ( n 2 ( n k m n k 0 n ( n ( n (m + k n k Proof The frst two equatons result from T m (n, k [x n ] + x ( k x mx + x [x n k ]( mx ( + x (k [x n k ] m x ( (k 0 0 [x n k ] ( k x ( m x + An alternatve proof for the secon entty f furnshe by usng the convoluton rule (rule 8

9 4KE - conv n [7] to get: [x n ] + x mx x k [x n k ] ( + x k mx ( + x k n k ( k + m n k 0 n k ( n 2 ( n k m, n k 0 whle the frst entty s obtane when we apply the convoluton rule n the symmetrc way, e, wth n k The thr equaton s a consequence of the factorzaton ( ( T m, + x (m + x, x snce ( (, +x has general term n n k ( n k Thus we have, for nstance, ( ( n n (m T m (n, k (m k k n k ( n ( n m k n Usng the stanar notaton for weght strbutons, we obtan, from equaton (8, ( n A (q T q (, mn ( ( n (q ( q mn mn 5 Applcatons to MDS coes We begn ths secton wth an example Example 9 The ual of the [7, 2, 6] Ree Solomon coe over GF (2 3 s an MDS [7, 5, 3] coe, also over GF (2 3 Thus the coe parameters of nterest to us are q 8, n 7, k 5 an mn n k + 3 Let D ag( ( ( 7 0, 7 (,, 7 7, 0, 0, enote the nfnte square matrx all of whose entres are zero except for those ncate We form the matrx prouct 9

10 (q DT q, wth q 8, to get {( } ag Column 3 (startng from column 0 of ths matrx then yels the weght strbuton of the [7, 5, 3] coe That s, we obtan the vector (, 0, 0, 245, 225, 5586, 2838, 2873, where we have mae the austment A 0 Thus we obtan A 0 A A 2 A 3 A 4 A 5 A 6 A We moreover notce that the numbers (0, 0, 0,, 5, 38, 262, 839, whch correspon to the ratos A /((q ( n, are elements of the sequence wth ornary generatng functon +x ( x 7x +x 3 x 3 (+x 4x 8x 2 20x 3 x 4 Hence they satsfy the recurrence a n 4a n + 8a n a n 3 + 7a n 4 Ths last result leas us to efne the weght ratos of a q-ary [n, k, ] MDS coe to be the ratos A /((q ( n We are now n a poston to summarze the results of ths paper Theorem 0 Let C be a q-ary [n, k, ] MDS coe The weght strbuton of C, auste for A 0, s obtane from the -th column of the matrx {( } ( n + x (q Dag (q x, x + x Moreover, the weght ratos of the coe satsfy a recurrence efne by the ornary generatng +x functon ( x (q x +x 0

11 Proof Inspecton of the expressons for the general term T q an the formulas for A yel the frst statement The secon statement s a stanar property of the columns of a Roran array Thus the weght ratos satsfy the recurrence a n ((q ( ( 0 an + ((q ( ( an ((q ( ( 2 an + (q a n Lettng R 0 0, an R A /((q ( n for > 0, we therefore have R l ( ((q 0 ( R l + where mn n k + In terms of the A, ths therefore gves us ( n (q ( ( + A ( n A 0 For nstance, we have 2873 ( [ ] an 2838 ( [ ] Acknowlegements The author woul lke to thank an anonymous revewer for ther careful reang of ths manuscrpt an for ther constructve remarks References [] D Merln, DG Rogers, R Sprugnol & MC Verr, On some alternatve characterzatons of Roran arrays, Canaan Journal of Mathematcs, 49 (2 (997, pp [2] F J MacWllams, N J A Sloane, The Theory of Error-Correctng Coes, North Hollan, Amsteram, NL, 2003 [3] W W Peterson, E J Welon, Error-Correctng Coes, 2n e MIT Press, Cambrge, US, 984 [4] L W Shapro, S Getu, W-J Woan an LC Wooson, The Roran Group, Dscr Appl Math 34 (99 pp

12 [5] N J A Sloane, The On-Lne Encyclopea of Integer Sequences Publshe electroncally at nas/sequences/,2007 [6] N J A Sloane, The On-Lne Encyclopea of Integer Sequences, Notces of the AMS, 50 (2003, [7] R Sprugnol, Combnatoral Ienttes, resp/goulbkpf/,2007 [8] R Sprugnol, Roran arrays an combnatoral sums, Dscrete Math,32 (994 pp [9] J H van Lnt, R M Wlson, A Course n Combnatorcs, 2n e Cambrge Unversty Press, Cambrge, 200 [0] E W Wessten, Mathematcs Subect Classfcaton: Prmary B83; Seconary 94B05, B37, B65 Concerns sequences A00738, A330, A33 2

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

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