On ucceive packing approach to multidimenional (M-D) interleaving Xi Min Zhang Yun Q. hi ankar Bau Abtract We propoe an interleaving cheme for multidimenional (M-D) interleaving. To achieved by uing a novel concept of bai interleaving array. A general method of obtaining a variety of bai interleaving array i preented. Baed on the bai interleaving array, we then propoe an interleaving technique, called ucceive packing, to generate the interleaved array of arbitrary ize. It i hown that the propoed technique can pread any error burt of m k 0 mk 1 within mn 0 mn 1 array (1 k n 1) effectively o that the error burt can be corrected with imple random error correcting-code (provided the error correcting-code i available). It i further hown that the technique i optimal for combating a et of arbitrarily-haped error burt. ince thi algorithm need to be implemented only once for a given M-D array, the computational cot i i low. Key word: Bai interleaving array, Multidimenional interleaving, error burt, randomerror-correction code 1 Introduction With rapid development of the information technology, two-dimenional (-D) and threedimenional (3-D) data handling i being widely ued. Application include -D and 3-D magnetic and optical data torage, charged-coupled device (CCD), -D barcode, and information hiding in digital image and video equence. Correcting error burt i an important problem in all of thee application. Thu, the iue of reliability of M-D information i an important tak, with both theoretical and practical ignificance. In thi paper, we addre the protection of multidimenional (M-D, M ) digital data. pecifically, we how how to pread the burt (cluter) of error in uch a way that they can be corrected by imple error correction code. ECE Dept, New Jerey Intitute of Technology, Newark, NJ 0710. ECE Dept, New Jerey Intitute of Technology, Newark, NJ 0710. Room A05, IBM T. J. Waton Reearch Center, New York 1053.
One-dimenional (1-D) interleaving technique ha been well documented in the literature (ee e.g., [1]). The main idea i to huffle the code ymbol from different codeword o that error burt encountered in the tranmiion are pread acro multiple codeword when the codeword are recontructed at the receiving end. Conequently, the error occurring within one codeword may be mall enough to be corrected by uing imple random-error-correction code. Extending thi trategy of one-dimenional (1-D) interleaving technique to the M- D ituation in order to combat error burt with ome random-error-correction code ha become the mot common approach to the correction of error burt. ome M-D interleaving technique for combating M-D error burt have been propoed in [, 3, 4, 5]. Among them, Almeida et.al. preent -D interleaving reult for circular-hape error burt []. Their reult cannot be generalized to non-circular haped burt or to higher dimenion. The United Parcel ervice (UP) combine the 1-D interleaving technique with a writing procedure to protect -D barcode [3]. However, thi approach cannot effectively pread the -D error burt [6, 7]. Adbel-Ghaffar [4] tudie ome theoretical apect of -D interleaving, but only preent unproven concept. A more comprehenive interleaving technique i dicued in [5], in which an error burt i defined a an arbitrarily-haped, connected area volume in the multidimenional pace. In thi method, for each burt ize t 0, a pecific algorithm i implemented, which can optimally correct arbitrarily-haped error burt of ize t 0. Furthermore, it i oberved that when the burt ize t increae, i.e., when t > t 0, the algorithm with a et of new parameter need to be implemented in order to correct the larger error burt of arbitrary hape. Likewie, when the burt ize decreae, i.e., t < t 0, the interleaving array that i optimal for burt ize t 0 i not optimal any more. ince, in practice, the ize of error burt are not known in advance, application of the technique i omewhat limited. By contrat, the ize of a given -D array (e.g., the ize of image and video frame) i known in many application. Motivated by thee obervation, a novel method, called ucceive packing (P), to -D interleaving, i propoed a a different and complementary technique [7, 8]and optimal performance of P on quare array of ize n n i proved. However, the analyi and application of P i retricted to quare array of ize n n. In thi paper, we firt propoe the novel concept of bai interleaving array. It characteritic are dicued and a method for it contruction i propoed. We then propoe to generate a large cla of interleaving array by ucceive packing of bai interleaving array. The performance of the propoed cheme in burt error correction i dicued. The paper i organized a follow. In ection, we introduce definition neceary for the remaining part of the paper. We then preent the concept of bai interleaving array for M-D interleaving, and the correponding -D method i hown to be optimal. Next, in ection 3, we propoe the P approach to M-D interleaving. We how that it work well in a et of error burt when ize of the -D array i given, but ize of error burt i not known in advance. Finally, concluion are drawn in ection 4.
Bai Interleaved Array Unle otherwie tated, for the ake brevity of dicuion, our preentation will be retricted to one-random-error-correction code. All reult can be extended to r-randomerror-correction code with r > 1 in a traightforward manner. The philoophy behind interleaving to combat burt of M-D error i imilar to that in the 1-D ituation. Looely peaking, with interleaving, the element in an M-D array are rearranged o that error in the interleaved M-D array are eparated a far away a poible from in the de-interleaved array. Error burt correction i, thu, facilitated if there i only one error in each codeword in the de-interleaved M-D array. Definition.1. Let C be an M-D code of m 0 m 1 m M 1 over GF(q). A codeword of C i an M-D array of m 0 m 1 m M 1, with each element of the M-D array aigned with a code ymbol. Note that GF (q) denote Galoi field over q element. The implet field i the binary field, GF () = {0, 1}. Definition.. In -D array, the neighbor of element (x, y) are denoted by (x + 1, y), (x 1, y), (x, y + 1), (x, y 1) In 3-D array, the neighbor of element (x,y,z) are denoted by (x + 1, y, z), (x 1, y, z), (x, y + 1, z), (x, y 1, z), (x, y, z + 1), (x, y, z 1) provided thoe element exit. Natural extenion of Definition. in higher dimenion apply. element ha M neighbor. For M-D array, an Definition.3. A burt i a ubet of the given M-D array B, in which any element ha at leat one neighbor contained in B. It ize i defined a the number of element in B. Definition.4. The ditance between any two element i the length of the hortet path between the two element. Here, path conit of a quence of neighbor connecting the two element. ince interleaving involve huffling code ymbol o that each element in an error burt i pread into a different codeword, if any two element within a ditinct codeword are eparated in the de-interleaved array uch that their ditance i maximized, then a large error burt can be hopefully corrected. Let A be an M-D array of ize m 0 m 1 m M 1. We re-index each element i0,i 1,,i M 1 of A a k with k being a function of i 0 to i M 1 (for intance, for -D array, we can have k = m 1 i 0 + i 1 ). Now conider a partitionin of A into L block with 1 L N, where N = m 0 m 1 m M 1. That i, each block o generated contain K = N/L element. 3
Definition.5. In the above cheme of partitioning followed by reindexing, any element having an index k with K(d 1) k < Kd, 1 d L i aid to belong to the d-th block. An element having index K(d 1) i referred to a the beginning element of the d-th block. All element belonging to the ame block are referred to a the K-equivalent element. According to Definition.5, we ee that l, l+1 are -equivalent element; 3l, 3l+1, 3l+ are 3-equivalent element. It i obviou that K 1 -equivalent element are alo K -equivalent element if K /K 1 i integer. Let one block be a codeword with length K, then all element of a ditinct codeword i K-equivalent with each other. Hence, the objective of effective interleaving i tranformed to the problem of maximizing the minimum ditance between any two K-equivalent element. If, for each k = 0 to M 1, m k i prime, then the number of codeword that the correponding M-D array can contain i an integer multiple of m n (n < M). Motivated by thi obervation, we propoe the concept of bai interleaving array next..1 quare -D bai array Definition.6. Conider a interleaving array B of ize m m, where m i prime. If the minimum ditance between any two m-equivalent element attain the maximum, then we call thi array a bai interleaving array. It i obviou from Definition.6 we have quare bai array of ize, 3 3, 5 5, etc. In [7, 8] an optimal interleaving technique baed on the ucceive packing of a pecific array i preented. In fact, thi particular array i an example of a bai interleaving array. Theorem.1. The -D array [ 0 3 1 ] (.1) i a bai interleaving array. Proof. In a array, the ditance from one corner to it oppoite corner i the maximum ditance between any two element. It i obviou that thi ditance equal to. For the two -equivalent element pair ( 0, 1 ) and (, 3 ), it i eaily een the ditance between 0 and 1, a well a the ditance between and 3 attain the value. Thu, Theorem.1 i proved. In order to contruct a bai interleaving array, it i neceary to know the upper bound of the minimum ditance. Note that the number of m-equivalent element of each element i m 1 in a quare m m array. Thu, we need to contitute a -D phere with ize m centered around each of the m-equivalent element. The m phere hould be able to tile to a m m array without overlapping. Then the maximum radiu of thi phere i the upper bound of the minimum ditance. Thi problem wa firt approached in [9] for m i odd. 4
Figure 1: Typical -D phere with ize, 5, 8, 13, 18. Later the idea wa extended in [5] to even m cae. It ha been proven that if m i even and m = t /, then the upper bound of the radiu i t; if m i odd and m = (t + 1)/, then the upper bound i alo t. ome example of -D phere are hown in Fig. 1. Notice that phere of ize m do not exit for value of m equal to 3, 4, 6, 7, 9, 10, 11,. Thu, the upper bound of the minimum ditance i the radiu of the larget phere with ize le than m. According to thi obervation, the upper bound i for m 4; the upper bound i 3 for 5 m 7; upper bound i 4 for 8 m 1, etc. In the following we preent a method of contructing quare bai interleaving array. Procedure.1. Let A be a -D array of ize m m (m ), and let d r be the upper bound of the ditance between any two element. We expre coordinate (i, j) of each element toroidally i.e., modulo integer m. We firt attribute the element of firt row a 0, m, m,, m(m 1), which are in the location (0, 0), (0, 1),, (0, m 1). Then, let X = 1 and Y = d r 1. For each element with location (i, j), add 1 to the ubcript of thi element and put it in the location (i+x, j+y). For example, 1 i put in the location (X, Y). Repeat thi procedure until all of the poition are occupied. Example.1.1: Conider the cae of array. According to [5], we have d r =. Thu, X = 1, Y = 1. Uing the above procedure with X = 1, Y = 1, we have contructed the array a in Fig.. It can be een that it i exactly ame with our bai array above. 0 3 1 Figure : bai interleaving array Example.1.: Conider the cae of 3 3 array. According to [5], we have d r =. Thu, 5
X = 1, Y = 1. Uing the above procedure with X = 1, Y = 1, we have contructed the array a in Fig. 3. 0 7 5 3 1 8 6 4 Figure 3: 3 3 bai interleaving array Example.1.3: Conider the cae of 5 5 array. According to [5], we have d r = 3. Thu X = 1, Y =. Uing the above procedure with X = 1, Y =, we have contructed the array a in Fig. 4. 0 16 7 3 5 3 14 1 1 19 10 1 17 8 4 15 6 13 4 0 11 18 9 Figure 4: 5 5 bai interleaving array Theorem.. If the integer m i prime, then the quare array contructed by Procedure.1 i a bai interleaving array. Proof. According to Procedure.1, we firt generate the poition for element 0, m, m,, m(m 1). Then we generate the poition of their m-equivalent element repectively. It can be een that the correponding ditance of two co-poitional m-equivalent element in each of the m-equivalent et i the ame. For example, the ditance between 0 and k i equal to the ditance between m and m+k, where k < m. Therefore, if we can prove Theorem. for the m-equivalent et beginning with 0, then Theorem. i proved. Baed on the ame reaoning, if we can prove that the ditance between 0 and any it m-equivalent element i greater than or equal to d r, then it hold for any k with 0 < k < m. We firt conider the cae when d r i odd. Due to toroidal labeling of the coordinate, the coordinate of k i (kx, ky mod m). The ditance between 0 and k i kx + ky mod m. Thu, the problem i tranformed to proving the following inequality: kx + ky mod m d r (.) If ky < m then we have ky mod m = ky. Thu, kx + ky mod m = k(x + Y ). ince X + Y = d r, it i obviou that k(x + Y ) > d r. Now, let u conider the cae that ky > m. 6
Then for integer l 1 and l we can write Uing the above equation, we can obtain ky = l 1 m + l. ince X = 1, Y = d r 1, we have k = l 1m + l. Y Hence, the inequality (.) become which i atified if kx + ky mod m = l 1m + l d r 1 + l. l 1 m + l d r 1 + l d r, (.3) l 1 m d r (l + 1)d r l. According to a reult in [5], we have m d r+1. Thu, if l 1 d r + 1 d r (l + 1)d r l hold, then inequality (.3) hold. The above inequality clearly hold for l 1 > 1. If l 1 = 1, then it follow that d r + 1 d r (l + 1)d r l. Thu, the value of l hould atify the following condition l > d r. In order to find the range of allowable value of l when l 1 = 1, let u decompoe m a Now, if we let k = dr+1 + 1 then m = d r + 1 = (d r 1) d r + 1 + 1 (.4) ky = (d r 1) d r + 1 + 1 + (d r ) = m + (d r ) (.5) Hence we get l = d r. According to the above procedure, d r i the minimum of l when l 1 = 1. Thu, the theorem i proved for the d r odd cae. imilar reaoning applie when d r i even. 7
In Procedure.1, we propoed a technique for generating the bai interleaving array. Next, we generalize thi method to any m m array uch that the minimum ditance between any two m-equivalent element attain the maximum value. Procedure.. Let A be a -d array of ize m m (m ), and let d r be the upper bound on the ditance between any two element of the array. We label the coordinate of the array toroidally on m i.e., we replace the corordinate (i, j) with their value modulo m. We firt attribute the element of firt row a 0, m, m,, m(m 1), which are in the location (0, 0), (0, 1),, (0, m 1). Then let X = 1, Y atify the condition d r 1 Y d r, with Y and m relatively prime. For each element with location (i, j), add 1 to the ubcript of the element and put it in the location (i+x, i+y). For example, 1 i put in the location (X, Y ). Repeat thi procedure until all of the poition are occupied. Theorem.3. For any integer m > 1, the quare array contructed by Procedure. attain the maximum in the ene of minimum ditance between any two m-equivalent element. Proof of Theorem.3 i imilar to proof of Theorem.. Notice that Contruction.1 in [5] i a pecial cae of Procedure., where Y = b r for m = b r+1, and Y = b r + 1 for m = b r.. Rectangular bai interleaving array Next, we generalize the reult of the previou ection to rectangular bai array. We have the following reult. Theorem.4. Let m n be a rectangular bai array. If m < n, then the upper bound of the minimum ditance between any two n-equivalent element i the ame a the minimum ditance of any two m-equivalent element in a m m bai array. If m > n, then the upper bound of the minimum ditance between any two m-equivalent element i the ame a the minimum ditance of any two n-equivalent element in a n n bai array. Proof. To prove Theorem.4, we firt prove that the upper bound of it minimum ditance cannot be greater than the correponding quare bai array. Then we how that the equality can be obtained. Let u aume for definitene that the minimum ditance i greater than the correponding quare bai array, and the minimum ditance of any two n-equivalent element greater than the minimum ditance of m m bai array. Let u then truncate the m n array to m m array. According to our aumption, the newly obtained m m array will have the minimum ditance larger than the m m bai interleaving array, which contradict the definition of bai interleaving array. Hence, the minimum ditance of the n-equivalent element in the m n array cannot be larger than the correponding m m quare bai interleaving array. To obtain the ame minimum ditance a the quare bai interleaving array, we can change Procedure.1 lightly to generate the rectangular bai interleaving array. 8
0 7 5 4 3 1 6 Figure 5: 3-D bai interleaving array. Procedure.3. Let A be a -D array of ize m n, and let d r be the upper bound of the ditance for the correponding quare interleaving array. If m < n, we re-expre i of row coordinate (i, j) toroidally a i mod m. We firt attribute the element of firt row a 0, n, n,, (m 1)n, which are in the location (0, 0), (0, 1),, (0, m 1). Then let X = d r 1, Y = 1. For each element with location (i, j), add 1 to the ubcript of the element and put it in the location (i+x, j+y). For example, 1 i put in the location (X, Y). Repeat thi procedure until all of the poition are occupied. If m > n, an analogou procedure i followed. It i eay to ee by uing argument a in the proof of Theorem., that Procedure.3 yield the ame minimum ditance a the correponding quare array..3 3-D bai interleaving array In thi ection, we attempt to briefly decribe via ome example how to extend the reult of the previou ection to higher dimenion. Definition.7. Conider a 3-D interleaving array B of ize l m n, where l, m, n are prime, and l m n. If the minimum ditance between any two mn-equivalent element attain the maximum, then we call thi array a bai interleaving array. For a quare 3-D array of ize m m m with minimum ditance d r, it ha been proven that m i bounded by m d3 r + d r, for d r even, 6 m d3 r + 5d r, for d r odd, 6 where m i the ize of 3-D phere [5]. Thi reult i further extended for M-D array in [5]. However, thi bound cannot be alway to achieve for M 3. For detail we refer to [9, 5]. Here, we preent an example of a 3-D bai interleaving array of ize, which will be ueful in our dicuion to follow. 9
3 ucceive Packing of Bai Interleaving Array The initial idea of ucceive packing for interleaving wa preented in [7, 8], where the author focu on the interleaving of array of ize n n. Whether it can applied to M- D array with arbitrary ize had not been invetigated. Alo, it i not clear if it optimal performance hold for rectangular -D array. In thi ection, we firt preent an M-D interleaving technique baed on the ucceive packing of bai interleaving array. Then it performance for preading error burt i analyzed. ubequently, it optimality i dicued and proved. 3.1 ucceive Packing Now we dicu the propoed P technique in M-D cae. Procedure 3.1. (M-D interleaving uing the ucceive packing) Conider an M-D bai interleaving array of ize m 0 m 1 m M m M 1. The interleaving array i the original bai interleaving array itelf. When m 0 = m 1 = m M 1 = 1, it i 1 = [ 0 ] (3.6) where 0 repreent the element in the array, and 1 the array. The ubcript in 1 repreent the total number of element in the interleaving array. Given interleaving array N of ize N = m 1 m m M m M 1, the interleaving array N can be generated by tranferring each element i in N to a M-D array according to the operation N N + i (thi operation i decribed further in the following). Thi packing procedure i carried out ucceively to generate N K by tranferring each element i in N K to a M-D array according to the operation N N K 1 + i. In the above procedure, the operation N N + i i the key point. Generalizing what we preented in [7], operation N N + i generate a M-D array with the ame dimenionality a N. Furthermore, each element in N N + i i indexed in uch a way that it ubcript equal to the N time of that of the correponding element (i.e., element occupying the ame poition in the M-D array) in N plu i. A few example are preented next. Example 3.1: Given a 1-D bai array 3 = { 0, 1, }, the interleaving array i 9 = { 0, 3, 6, 1, 4, 7,, 5, 8 }. Example 3.: Given a 3 3 bai array a in Fig.3, the 9 9 interleaving array i generated a in Fig.6. Example 3.3: Given a bai array a in Fig.5, the 4 4 4 interleaving array i generated a in Fig.7, wherea the left hand ide diplay array obtained via the operation 8 8 + 5. To generate a interleaving array with arbitrary ize, we ue the ucceive packing method baed on a combination of different bai interleaving array. For intance, given bai 10
ƒ ƒ Œ Œ Š Š Ž Ž ˆ ˆ RQ XW BA HG 1 87 rq xw " "!! (' ba hg 0 7 54 63 9 36 45 7 18 7 70 5 5 68 50 34 16 79 3 14 77 4 6 1 8 55 5 59 41 3 3 30 57 66 1 39 48 75 1 64 10 37 46 73 19 8 35 6 71 17 44 53 80 6 6 33 60 69 15 51 78 4 4 31 58 67 13 40 4 49 76 9 56 65 11 38 47 74 0 Figure 6: ucceive Packing generated 9 9 interleaving array. 8x +0 8x +7 - -.. 8 - -.. 8 0 61 # #$ %& ) ) * * 45 13 1 37 9 53 ] ] ^ ^ 8x +5 C CD EF K KL 8 M M N N I I J J O O P P + +, 3 3 4 4 56 ; ; < < 8x +4 = => 9 9:??@ / / 0 0 c cd ef i i 8 j j 8 } }~ 8x + k kl 8 m m n n 8x +3 o o p p T UV t t uv Y Y Z Z y yz [ [\ _ _ ` ` 8x +1 { { 8x +6 8 8 Figure 7: ucceive Packing generated 4 4 4 interleaving array. interleaving array N and, we can generate the interleaving array N by { N + 0, N + 1}. Example 3.4: Given bai array a in Fig. and 3 3 bai array a in Fig.3, the 6 6 interleaving array i generated a in Fig.8. 3. Performance Analyi Before embarking on performance analyi of our P baed M-D interleaving technique, we introduce the following definition. Definition 3.1. Conider two burt B 1 and B having the ame ize and hape in an interleaving M-D array. If each element in a burt (e.g., B 1 ) i either an element of another burt (e.g., B ), or K-equivalent of an element of another burt (B ), then we ay that burt B 1 and B are K-equivalent burt. In the remainder of the paper, when dicuing error burt correction, we may conider 11
4x + 0 4x + 9 9 0 1 4 8 4 16 0 3 8 3 15 7 31 7 19 3 35 11 14 6 30 6 18 34 10 1 13 5 9 5 17 1 33 9 4x + 3 4x + 1 9 9 Figure 8: ucceive Packing generated 6 6 interleaving array. each et of equivalent element defined in Definition.5 a a M-D codeword. Thi implie that a codeword conit of a et of conecutive code ymbol. Thi i neceary ince we need to dicriminate code ymbol within a codeword in our enuing dicuion of the P technique for M-D interleaving. An error burt (in the interleaved array) i aid to be pread, and can be corrected with one-random-error-correction code, if each element in the burt i pread in ditinct codeword of the de-interleaved array. From thi point of view, it i eay to ee that given two equivalent burt, if one i interleaved then the other burt mut have alo been interleaved. We may now tate the following reult. Lemma 3.1. Let A be a -D array of ize m n 0 m n 1 obtained by uing ucceive packing of a bai interleaving array of ize m 0 m 1. Then all burt of ize m k 0 mk 1 in A with k < n are K 1 -equivalent, where K 1 = (m 0 m 1 ) n k. The proof of thi lemma i ommitted for brevity. Now we are in a poition to preent the following theorem. Theorem 3.1. Conider a -D array A of ize m n 0 m n 1. Then any burt of ize m m 0 m m 1 with m n in the interleaving array A obtained by uing the ucceive packing i pread in the de-interleaved array o that each element of the burt fall into a ditinct block of ize m0 n m m1 n m. Theorem 3.1 indicate that, if a ditinct code ymbol i aigned to each element in a block (cf. to Definition.5), and all the code ymbol aociated with an individual K-equivalent cla form a ditinct codeword, then burt error correction with a one-random-error-correction code i guaranteed, provided the code i available. Furthermore, the interleaving degree equal the ize of the burt error, hence minimizing the number of codeword required in an interleaving cheme. In other word, with the ucceive packing technique, the interleaving degree attain the lower bound (the interleaving gain) 1. In thi ene, the ucceive packing 1 ee [8] for definition of interleaving degree and interleaving gain. 1
interleaving technique i optimal. Note that dicuion in [8] can be conidered to be a pecial cae of Theorem 3.1 with m 0 =, m 1 =. We conjecture that following reult hold. Conjecture 3.1. Conider an M-D array A of ize m n 0 m n 1 m n M 1. Then any burt of ize m m 0 mm 1 mm M 1 with m n in the interleaving array A obtained by uing the ucceive packing i pread in the de-interleaved array o that each element of the burt fall into a ditinct block of ize m0 n m m1 n m m n m M 1. In ection 3.1, we propoed to generate arbitrary ize interleaved array by combining different bai interleaving array. Here, we firt how how to generate a quare -D array of ize m m. We then prove it optimal performance. The procedure can be decribed a follow. Procedure 3.. Generate the m m interleaving array according to our P technique. Generate the m m interleaving array a follow [ ] 4 m + 0 4 4m = m + 4 m + 3 4 m + 1 (3.7) Let l i,j denote the ubcript of the correponding element in the -D interleaving of array 4m of ize m m. The m m interleaving array i generated a m 4m + l 0,0 m 4m + l 0,m 1 4m 4 =.. (3.8) m 4m + l m 1,0 m 4m + l m 1,m 1 We firt need the following lemma. Lemma 3.. Let C be a cluter of ize m in a -D array of ize m 1 m 1, where m < m 1. Then there mut exit a rectangular block R 1 of ize m m, and/or a rectangular block R of ize m m uch that C i entirely contained in either R 1 or R, or in both. Proof. For the purpoe of etablihing a contradiction, let u hypotheize that there do not exit block R 1 and R a in the tatement of the Lemma entirely containing C. Then, C would be outide of R 1 either in the X, or in the Y direction. ince the length of R 1 in Y direction i m, which i equal to the ize of C, it i only poible for C to be outide of R 1 in X direction. Hence we have C X > m, where the C X i the dimenion of C along X direction. ince C i not entirely contained in R, baed on the ame reaoning above, we would have C Y > m, where C Y i the dimenion of C along Y direction. Our hypothei then would imply that the ize ize(c) of cluter C, atify the following: ize(c) C X + C Y 1 (m + 1) 1 = m + 1, which contradict that C i of ize m. The lemma i hence proved. 13
Theorem 3.. The -D interleaving array generated by Procedure 3. i optimal in the ene that it can pread arbitrary burt of ize m to ditinct codeword of ize m. Proof. According to Lemma 3.1 and Theorem 3.1, it i eay to ee that any two m m burt within the generated m m array are m -equivalent, and any two m m or m m burt within the generated m m array are m -equivalent. Thu, any burt with ize m m or m m can be pread into ditinct block with ize m. However, according to Lemma 3., an arbitrary burt of ize m i necearily contained in a burt of ize either m m or m m. Theorem 3. i, thu, proved. Theorem 3. indicate that if a ditinct code ymbol i aigned to each element in block of ize m (refer to Definition.5) and all the code ymbol aociated with a block form a ditinct codeword, then the P technique can correct arbitrarily-haped error burt of ize m with one-random-error-correction code, provided the code i available. That i, the P technique achieve the ame performance a that achieved by the technique in [5]. Comment: Needle to ay that there are certain contraint with the P technique. Namely, it i not guaranteed that the upper bound of the minimum ditance can be achieved for the larger -D array (multiple bai interleaving array) with ize (m + 1) (m + 1). However, ince the P baed interleaving method i optimal for a large et of burt, it provide a veratile tool for burt error correction. In ummary, the P approach doe provide an effective way for M-D interleaving. For a given -D array of ize m n 0 m n 1, it can be applied once, and i optimal for a et of error burt having different ize defined in Theorem 3.1. In addition, for the cae of arbitrarily-haped error burt having a ize of m, to which both the P technique and the technique in [5] can be applied, the P approach can alo pread and correct arbitrarily-haped error burt with the ame lower bound obtained by uing the approach in [5]. For the bai interleaving array, we propoed a method which i proved to be optimal in -D cae. For M > D cae, thi optimality cannot guaranteed [9, 5]. 4 ummary In thi paper, we focu on how to realize effective M-D interleaving. We firt preent a novel concept, bai interleaving array. Baed on thi, a new interleaving method, called ucceive packing (P), i propoed to combat M-D error burt. We have proved that the propoed method can pread any error burt of m k 0 mk 1 (with 1 k n 1) to different code block in the array of m n 0 mn 1. Thu the imple error correction code which i optimal for independent channel can be ued to correct thi kind of error burt. It need to be implemented only once for a given M-D array, and i thereafter optimal for the et of error burt having different ize. 14
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