Floor Scale Modulo Lifting for QC-LDPC codes

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1 Floor Scale Modulo Lifting for QC-LDPC codes Niita Polyansii, Vasiliy Usatyu, and Ilya Vorobyev Huawei Technologies Co., Moscow, Russia arxiv: v2 [cs.it] 4 Feb 207 Abstract In the given paper we present a novel approach for constructing a QC-LDPC code of smaller length by lifting a given QC-LDPC code. The proposed method can be considered as a generalization of floor lifting. Also we prove several probabilistic statements concerning a theoretical improvement of the method with respect to the number of small cycles. Maing some offline calculation of scale parameter it is possible to construct a sequence of QC-LDPC codes with different circulant sizes generated from a single exponent matrix using only floor and scale operations. The only parameter we store in memory is a constant needed for scaling. Keywords: QC-LDPC code, floor lifting, modulo lifting, bloc cycle, girth. I. INTRODUCTION Low-density parity-chec (LDPC codes were first discovered by Gallager [], generalized by Tanner [2], Wibberg [3] and rediscovered by MacKay et al. [4], [5] and Sipser et al. [6]. Quasi-cyclic low-density parity-chec (QC-LDPC codes are of great interest to researchers [7] [] since they can be encoded and decoded with low complexity and allow to reach high throughput using linear-feedbac shift register [2] [4]. One advantage of QC-LDPC codes based on circulant permutation matrices (CPM is that it is easier to analyze their code and graph properties than in the case of random LDPC codes. The performance of LDPC codes is strongly affected by their graph properties such as the length of the shortest cycle, i.e., girth [5], [6], and trapping sets [7], [8] and code properties, e.g., the distance of the code [9], [20] and the ensemble weight enumerator [2]. The main contribution of the paper is a novel approach for constructing a quasi-cyclic LDPC code of smaller length by lifting a given QC-LDPC code. The proposed method can be considered as a generalization of floor lifting method introduced in [22], [23]. Maing some offline calculation it is possible to construct a sequence of QC-LDPC codes with different circulant sizes generated from a single exponent matrix of QC-LDPC code having the largest length. The only parameter we store in memory is a constant needed for scaling in the lifting procedure. The outline of the paper is as follows. In Section II, we introduce some basic definitions and notations for our presentation. In Section III, we review state-of-art lifting methods for QC-LDPC codes. Also assuming some natural assumption we prove some probabilistic statements with respect to cycles of length 4 and provide a comparison between lifting procedures. In Section IV, we present our floor scale modulo lifting method for QC-LDPC codes and prove several probabilistic statements concerning theoretical improvement of the method with respect to the number of small cycles. The performance of QC-LDPC codes obtained by the floor scale modulo lifting method is investigated by simulations in Section V. II. QC-LDPC CODES A QC-LDPC code is described by a parity-chec matrix H which consists of square blocs which could be either zero matrix or circulant permutation matrices. Let P (P ij be the L L circulant permutation matrix defined by, if i + j mod L P ij 0, otherwise. Then P is the circulant permutation matrix (CPM which shifts the identity matrix I to the right by i times for any, 0 L. For simplicity of notation denote the zero matrix by P. Denote the set, 0,,..., L } by A L. Let the matrix H of size ml nl be defined in the following manner P a P a2 P an P a2 P a22 P a2n H......, ( P am P am2 P amn a i,j A L. Further we call L the circulant size of H. In what follows a code C with parity-chec matrix H will be referred to as a QC-LDPC code. Let E(H (E ij (H be the exponent matrix of H given by: a a 2 a n a 2 a 22 a 2n E(H......, (2 a m a m2 a mn i.e., the entry E ij (H a ij. The mother matrix M(H is a m n binary matrix obtained from replacing s and other integers by 0 and, respectively, in E(H. If there is a cycle of length 2l in the Tanner graph of M(H, it is called a bloc-cycle of length 2l. Any bloc-cycle in M(H of length 2l corresponds both to the sequence of 2l CPM s P a, P a2..., P a 2l } in H and sequence of 2l integers a, a 2... a 2l } in E(H which will be called exponent chain. The following well nown result gives the easy way to find cycles in the Tanner graph of parity-chec matrix H. Proposition. [5]. An exponent chain forms a cycle in the Tanner graph of H iff the following condition holds 2l i ( i a i 0 mod L.

2 III. LIFTING OF QC-LDPC CODES A. State-of-art Lifting Methods Consider a QC-LDPC code with ml 0 nl 0 parity-chec matrix H 0 with circulant size L 0, m n exponent matrix E(H 0 (E ij (H 0 and mother matrix M(H 0. Given a set of circulant sizes L }, L < L 0, lifting is a method of constructing QC-LDPC codes with ml nl parity-chec matrices H from H 0, which have the same mother matrix M(H M(H 0 and entries of exponent matrices E(H satisfy E ij (H L. Therefore, it suffices to specify a formula using which we recalculate each value of E(H from E(H 0. In paper [22] two lifting approaches are given. Floor lifting is defined as follows: L L E ij (H 0 E ij (H 0, if E ij (H 0, (3, otherwise. Modulo lifting is determined by the following equation: E ij (H 0 mod L, if E ij (H 0, E ij (H (4, otherwise. Now we prove several probabilistic statements. Consider an exponent chain of length 4 with exponent values a, b, c, d [ ] a b A, c d each element is chosen independently and equiprobable from the set 0,,..., 2q }, L 0 2q is a circulant size, q > 2. Notice that the probability of the event C 0 : the exponent chain with exponent values a, b, c and d forms a cycle, i.e., a b c + d 0 mod 2q, is equal to /(2q. Assume that we use some lifting method to obtain exponent values a, b, c, d [ ] a b B c d, for circulant size L q. We are interested in the probabilities of an event C : the exponent chain with exponent values a, b, c and d forms a cycle given the event C 0 and given the event C 0. In Sections III-B and III-C we obtain these probabilities for floor lifting and modulo lifting, respectively. Finally, we summarize results and compare these two methods in Section III-D. B. Floor Lifting Let a 2a +a 2, b 2b +b 2, c 2c +c 2 and d 2d + d 2, a 2, b 2, c 2, d 2 0, }. One can see that a a, b b, c c, d d. Given the event C 0 occurs, i.e. 2(a b c + d + (a 2 b 2 c 2 + d 2 0 mod 2q. the event C, i.e., a b + d c 0 mod q, is equivalent to the condition a 2 b 2 c 2 + d 2 0. From C 0 it follows that a 2 b 2 + d 2 c 2 0 mod 2. Therefore, the conditional probability Pr(C C 0 Pr(a 2 b 2 c 2 + d 2 0 C 0 Pr(a 2 b 2 c 2 +d 2 0 a 2 b 2 c 2 +d 2 0 mod 2 3/4. Indeed we have exactly equiprobable choices for a 2, b 2, c 2, d 2 depicted in Table I, 6 ( 4 2 of which give the cycle. TABLE I POSSIBLE CHOICES FOR a 2, b 2, c 2, d 2 a 2 b 2 c 2 d 2 a 2 b 2 c 2 +d Now let us find the probability Pr(C C 0. Since Pr(C C 0 Pr(C C 0 Pr(C 0 and Pr(C 0 2q 2q, it suffices to obtain Pr(C C 0. Find the number of all 4-tuples (a, b, c, d, such that a b c +d 0 mod q and a 2 b 2 c 2 + d 2 0 mod 2q. We have q 3 ways to choose a, b, c, d and 0 ways to choose a 2, b 2, c 2, d 2 for q > 2. Therefore, Pr(C C 0 0q3 (2q 4 5 8q and Pr(C C 0 5 4(2q. Let us sum up the results in Proposition 2. An exponent chain in E(H of length 4, which forms a cycle in the parity-chec matrix H with circulant size 2q, turns into a cycle in the parity-chec matrix H with circulant size q obtained after floor lifting with probability 3/4, while an exponent chain of length 4, which does not form a cycle, turns into a cycle with probability p fl 5/(4(2q. C. Modulo Lifting Let a a q +a 2, b b q +b 2, c c q +c 2, d d q +d 2, a 2, b 2, c 2, d 2 0,,..., q }. It is easy to chec that a a 2, b b 2, c c 2, d d 2. Given the event C 0 occurs, we have q(a b c + d + (a 2 b 2 c 2 + d 2 0 mod 2q. It follows that a b c + d a 2 b 2 c 2 + d 2 0 mod q, thus the conditional probability Pr(C C 0. Let us obtain probability Pr(C C 0. Since Pr(C C 0 Pr(C C 0 Pr(C 0

3 and Pr(C 0 2q 2q, we need to find Pr(C C 0. Calculate the number of all 4-tuples (a, b, c, d, such that a 2 b 2 c 2 +d 2 0 mod q and a b c + d 0 mod 2q. We have q 3 ways to choose a 2, b 2, c 2, d 2 and 8 ways to choose a 2, b 2, c 2, d 2 for q > 2. Therefore, Pr(C C 0 8q3 (2q 4 2q and Pr(C C 0 2q. As a result we have obtained the following Proposition 3. An exponent chain in E(H of length 4, which forms a cycle in the parity-chec matrix H with circulant size 2q, turns into a cycle in the parity-chec matrix H with circulant size q obtained after modulo lifting with probability, while an exponent chain of length 4, which does not form a cycle, turns into a cycle with probability p mod /(2q. D. Comparison Now summarize the results from Sections III-B and III-C in the following Theorem. Suppose that in exponent matrix E(H with circulant size 2q we have y exponent chains of length 4, which do not form a cycle, and x exponent chains of length 4, which form a cycle. Then mathematical expectations EC fl (EC mod of the number of cycles after floor lifting (modulo lifting for circulant size q are as follows: EC fl 3 4 x + 5 4(2q y, EC mod x + (2q y. Note that EC fl EC mod when y (2q x. Since usually we try to eliminate short cycles in matrix E(H, the number y is liely to be much greater than (2q x. So, we can conclude that modulo lifting is better than floor lifting with respect to the number of short cycles. IV. FLOOR SCALE MODULO LIFTING OF QC-LDPC CODES Now we introduce the proposed lifting method which we call floor scale modulo lifting:, Eij (H 0, E ij (H L L 0 ((r E ij (H 0 mod L 0, otherwise, (5 special parameter r is called a scale value. Define A(r: [ ] a(r b(r A(r, c(r d(r a(r ra mod 2q, b(r rb mod 2q, c(r rc mod 2q, d(r rd mod 2q. By C 0 (r denote the event: the exponent chain with exponent values a(r, b(r, c(r and d(r forms a cycle. Notice that for r coprime with 2q, i.e. (r, 2q, elements of matrix A(r have the same distribution as matrix A. Moreover, exponent chains from matrices A and A(r form a cycle simultaneously. Let a 2a + a 2, b 2b + b 2, c 2c + c 2 and d 2d + d 2, a 2, b 2, c 2, d 2 0, }. Suppose we use floor scale modulo lifting for L q with scale value r 2t +, 0 < r < 2q, which is coprime with 2q. Then we obtain matrix B(r: [ ] a B(r (r b (r c (r d, (r a 2a r + a 2 r (r a r + a 2 t mod q. 2 Other values b (r, c (r and d (r are represented in the same way. By C (r denote the event: the exponent chain with exponent values a (r, b (r, c (r and d (r forms a cycle. One can see that Moreover Pr(C (r C 0 Pr(C ( C C (r C 0 C ( C 0. Proposition 4. Let r, r 2 be two distinct integers, such that 0 < r, r 2 < 2q, (r, 2q, (r 2, 2q and r r 2 (q+ mod 2q. Then Pr(C (r C (r 2 C 0 0. In other words, for any scale values r and r 2 fulfilled the condition of Proposition 4 if the start exponent chain in the matrix A does not form a cycle then at least one exponent chain in the matrices B(r and B(r 2 does not form a cycle too. Proof: Let u be such integer that u r mod 2q. Note that C 0 (u C 0. Therefore, Pr(C (r C (r 2 C 0 Pr(C (r u C (r 2 u C 0 0. Assume events C (r u and C (r 2 u occur. Thus, a b c + d 0 mod q and (a b c + d r 2 + (a 2 b 2 c 2 + d 2 t 2 0 mod q, From + 2t 2 r 2 r 2 u mod q, 0 < r 2 < 2q. (a 2 b 2 c 2 + d 2 t 2 0 mod q, (a 2 b 2 c 2 + d 2 [ 2, 2], t 2 [, q ] and 2t 2 + q + it follows that a 2 b 2 c 2 + d 2 0. Hence (a b c + d 0 mod q and 2(a b c + d + (a 2 b 2 c 2 + d 2 a b c + d 0 mod 2q, i.e., we prove that C (r C (r 2 C 0. Remar. Note that if r r 2 (q + mod 2q, then r 2 r (q + mod 2q. Therefore, we can choose a set R of scale values of cardinality ϕ(2q/2 (ϕ(n is Euler s totient

4 function for even q and ϕ(2q for odd q, such that for every r, r 2 R the conditions of Proposition 4 are fulfilled. Consider a floor scale modulo lifting with a family R r, r 2,..., r Nr } of N r scale values, such that for any two scale values r i, r j R the conditions of Proposition 4 are satisfied. Let D (D ij be an N r y matrix, the i- th row corresponds to scale values r i R, and each column corresponds to one exponent chain of length 4 in E(H. We set D ij to if the j-th exponent chain forms a cycle after floor scale modulo lifting with scale value r i, and to 0 otherwise. The first x columns, which corresponds to cycles in exponent matrix with circulant size 2q, equal to the column of ones with probability 3/4 and to the column of zeros with probability /4. The rest y columns equal to the column of zeros with probability N r p fl 5N r 4(2q and to the column of weight with one at position i with probability p fl 5/(4(2q for each i [, N r ]. Let X i be equal to the number of ones in the i-th row. We are interested in the minimum number of cycles min(x, X 2,..., X Nr. For further calculations we assume that all columns of matrix D are chosen independently. Under this assumption exact formulas for the mathematical expectation EC fsml (N r 3x/4 + E min(x, X 2,..., X Nr could be easily written out, but they rather messy. We provide only formula for the case N r 2 in the form of Proposition 5. [28]. Suppose we have an exponent matrix E(H with circulant size 2q having x exponent chains of length 4, which form a cycle in H, and y exponent chains of length 4, which do not form a cycle. Then the mathematical expectation EC fsml (2 of the number of cycles of length 4 in the parity-chec matrix of circulant size q obtained after floor scale modulo lifting with N r 2 scale values, which satisfies the conditions of Proposition 4, is described by the following expression ( ( EC fsml (2 3 y n 4 x + n n n n0 ( y (2p fl n ( 2p fl y n. n The proof of Proposition 5 is provided in the full version of the given paper [28]. If y the asymptotic behavior of EC fsml (N r is given by Theorem 2. [28]. The mathematical expectation of the number of cycles of length 4 after floor scale modulo lifting has the following asymptotic form EC fsml 3 4 x + p fly c Nr y + o( y, if y, c Nr does not depend on y. Let us consider another scenario. Suppose that the number of cycles of length 4 in matrix H with circulant size L 0 2q is equal to 0, and the number y of exponent chains is fixed. Now we are interested in the probability that after lifting for the circulant size L q we will not obtain any cycle of length 4. We again assume that all events C are independent for all exponent chains, i.e., all columns of matrix D are chosen independently. Theorem 3. [28]. The probability of the absence of cycles of length 4 in the parity-chec matrix with circulant size q obtained after modulo lifting, floor lifting and floor scale modulo lifting is as follows P mod ( p mod y yp mod + O(q 2, P fl ( p fl y yp fl + O(q 2, N r ( P fsml (N r ( Nr ( p fl y q q O(q Nr, if y N r, q,, if y < N r, q. In this case we see that floor scale modulo lifting is much better than modulo and floor lifting. Table II shows one of possible advantages of the proposed lifting approach. We compare the floor lifting length adaption of QC-LDPC codes used in IEEE for rate /2 with the proposed floor scale modulo lifting. We apply the lifting methods to the 2 24 mother matrix. We have found optimal r scale value for our lifting approach with respect to girth and number of exponent chains which form cycles of the minimal length. In Table II for each circulant size the optimal r scale value, girth and the number of cycles are depicted. Note that the QC-LDPC code of IEEE standard was optimized with considering floor lifting method. If the QC-LDPC code with the maximal length size is not optimized with considering floor or modulo lifting method, then the superiority of the proposed floor scale modulo lifting will be more conspicuous. V. SIMULATION RESULTS QC-LDPC codes of smaller lengths can be obtained by lifting exponent matrix of QC-LDPC codes of maximal length. Their performance over an AWGN channel with BPSK modulation was analyzed by computer simulations. Figure shows the frame error rate (FER performance of rate 4 over 5 AR4JA code defined by protograph of size 3 from [24]. We use native lifting for fixed circulant sizes 6, 32, 64, 28} and floor modulo scale lifting beginning from parity-chec matrix H of circulant size 28 which goes down to circulant sizes 6, 32, 64}. BP decoder with 00 iterations is used. Figure 2 shows the SNR required to achieve 0 2 FER performance over an AWGN channel with QPSK modulation for 3 families of QC-LDPC codes with rate 8 over 9. Families A [26] and B [25] are the industrial state-of-art QC-LDPC codes with their own lifting. For family C we applied floor modular scale lifting. Layered normalized offset min-sum decoder with 5 iterations was used in simulations. Normalize

5 TABLE II G IRTH AND THE NUMBER OF SHORT CYCLES FOR FLOOR LIFTING AND FLOOR SCALE MODULO LIFTING Floor E scale modulo lifting r girth / cycles 95 6 / 3 4/ 6 / /4 9 6/4 2 6/3 4 6/3 3 6/4 Floor lifting, r E girth / cycles 24 6 / / 32 6 / 36 6 / / /9 68 6/8 72 6/ / /8 96 Fig.. Performance of AR4JA codes with different CPM sizes by the lifting L28, 256, 52. BP 00 it. and offset factors were optimized to improve waterfall performance [27]. In summary, the proposed lifting scheme supports fine granularity and avoids catastrophic cases for different lengths. R EFERENCES [] R. G. Gallager, Low-density parity-chec codes, IRE Trans. Inform.Theory, vol. IT-8, pp. 2-28, Jan Fig. 2. SNR required to achieve 0 2 FER [2] Tanner R. M. A Recursive Approach to Low Complexity Codes,IEEE Trans. Inform. Theory, IT-27, pp , September 98. [3] Wiberg, N. Codes and decoding on general graphs, Ph. D. dissertation, Linping University, 996. [4] D. J. C. MacKay and R. M. Neal, Near Shannon limit performance of low-density parity-chec codes, Electron. Lett., vol. 32, pp , Aug [5] D. J. C. MacKay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory, vol. 45, pp , Mar [6] M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inform. Theory, vol. 42, pp , Nov [7] Y. Kou, S. Lin, and M. P. C. Fossorier, Low-density parity-chec codes based on finite geometries: a rediscovery and new results, IEEE Trans. Inf. Theory, vol. 47, no. 7, pp , Nov [8] I. Djurdjevic, J. Xu, K. Abdel-Ghaffar, S. Lin, A class of low-density parity-chec codes constructed based on Reed-Solomon codes with two information symbols, Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes, Springer Berlin Heidelberg, [9] R. M. Tanner, D. Sridhara, A. Sridharan, T. E. Fuja, and D. J. Costello, Jr., LDPC bloc and convolutional codes based on circulant matrices, IEEE Trans. Inf. Theory, vol. 50, no. 2, pp , Dec [0] L. Zhang, Q. Huang, S. Lin, K. Abdel-Ghaffar, and I. F. Blae, Quasicyclic LDPC codes: An algebraic construction, ran analysis, and codes on latin squares, IEEE Trans. Comm., vol. 58, no., pp , Nov [] D. Divsalar, S. Dolinar and C. Jones, Construction of Protograph LDPC Codes with Linear Minimum Distance, 2006 IEEE International Symposium on Information Theory, Seattle, WA, 2006, pp [2] S. Myung, K. Yang, and J. Kim, Quasi-cyclic LDPC codes for fast encoding, IEEE Trans. Inf. Theory, vol. 5, no. 8, pp , 2005 [3] J. Kim, A. Ramamoorthy and S. W. Mclaughlin, The design of efficiently-encodable rate-compatible LDPC codes, IEEE Trans. Comm., vol. 57, no. 2, pp , February [4] Advanced hardware design for error correcting codes /Cyrille Chavet, Philippe Coussy,editors. Cham:Springer, 205., pp. 7-3 [5] M. P. C. Fossorier, Quasi-cyclic low-density parity-chec codes from circulant permutation matrices, IEEE Trans. Inf. Theory, vol. 50, no. 8, pp , [6] Y. Wang, S. C. Draper and J. S. Yedidia, Hierarchical and High-Girth QC LDPC Codes, IEEE Trans. Inf. Theory, vol. 59, no. 7, pp , July 203. [7] B. Vasi, S. K. Chilappagari, D. V. Nguyen and S. K. Planjery, Trapping set ontology, th Annual Allerton Conference on Communication, Control, and Computing (Allerton, Monticello, IL, 2009, pp. -7. [8] M. Diouf, D. Declercq, S. Ouya and B. Vasic, A PEG-lie LDPC code design avoiding short trapping sets, Proc. IEEE Inter. Symp. Information Theory, Hong Kong, 205, pp [9] R. Smarandache, P. O. Vontobel, Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds, in IEEE Trans. Inf. Theory, vol. 58, no. 2, pp , Feb [20] B. K. Butler and P. H. Siegel, Bounds on the Minimum Distance of Punctured Quasi-Cyclic LDPC Codes, IEEE Trans. Inf. Theory, vol. 59, no. 7, pp , July 203. [2] Dariush Divsalar, Ensemble Weight Enumerators for Protograph LDPC Codes, Proc. IEEE Inter. Symp. Information Theory,2006. pp , [22] Myung S., Yang K. Extension of quasi-cyclic LDPC codes by lifting. Proc. IEEE Inter. Symp. Information Theory, ISIT 2005, [23] Myung, S., Yang, K., and Kim, Y.. Lifting methods for quasi-cyclic LDPC codes. IEEE Comm. Lett., 2006, 0(6, [24] D. Divsalar, S. Dolinar and C. Jones, Construction of Protograph LDPC Codes with Linear Minimum Distance, Proc. IEEE Inter. Symp. Information Theory, ISIT 2006., Seattle, WA, 2006, pp [25] R-66370, LDPC rate compatible design, Qualcomm from 3GPP TSGRAN WG 86, 22th 26th August 206, Gothenburg, Sweden [26] R-67889, Design of Flexible LDPC Codes, Samsung from 3GPP TSG RAN WG 86, 22th26th August 206, Gothenburg, Sweden [27] Chen J., Tanner R. M., Jones C., Li Y. Improved min-sum decoding algorithms for irregular LDPC codes, Proc. IEEE Inter. Symp. Information Theory, ISIT 2005., Adelaide, SA, 2005, pp [28] Poliansii N., Usatyu V., Vorobyev I., Floor Scale Modulo Lifting,

6 A. Proof of Proposition 5 APPENDIX Proof: Consider a random variable min(x, X 2. Then E min(x, X 2 E E(min(X, X 2 X + X 2 y E min(y, n Y Pr(X + X 2 n, n0 Y B(n, 2. From and E min(y, n Y n 2 ( ( n n/2 2 n ( y Pr(X + X 2 n (2p fl n ( 2p fl y n, n it follows the statement of the proposition. B. Proof of Theorem 2 Proof: Consider a random vector X Nr+ y Nr X (X, X 2,..., X Nr, X Nr+, i X i. One can see that X has a multinomial distribution with y! Pr(x,..., x Nr+ x!... x p i0 fl ( N r p fl x Nr +. Nr+! By the Central limit theorem the distribution of random vector X EX y tends to normal distribution N (0, Σ as y, Σ is the covariance matrix of X. Let us prove that E min(x, X 2,..., X Nr E min X o( y as y. E min(x, X 2,..., X Nr E min X Nr yp(min X X Nr+ yp(x Nr+ y/(n r + ( yp X Nr+ EX Nr+ (6 ( p fl N r (N r + From N r ϕ(2q q it follows that p fl N r > p fl for q > 2. The following chain of inequalities taes place p fl N r > p fl Denote Chernoff inequality p fl (N r + N r N r < N r + <. (7 ( p fl N r (N r + ( p fl N r(n r+ as δ for some δ > 0 and use the yp (X Nr+ EX Nr+( δ ye δ 2 EXNr + 2 o( y. x i Let denote as min n (X the function which equals max(min(min(x, n, n. This function is continuous and bounded, hence lim E min y ( X EX y lim lim E min X EX n n y y lim n E min nn (0, Σ c Nr E max N (0, Σ > 0. Therefore, E min N (0, Σ c Nr, (8 E min(x, X 2,..., X Nr E min X + o( y min EX yc Nr + o( y p fl y yc Nr + o( y. (9 C. Proof of Theorem 3 Proof: We prove the formula for P fsml only. Firstly, find the probability P that the first rows of the matrix D contain only zeros P ( p fl y. Secondly, using the inclusion-exclusion principle we get N r ( P fsml ( Nr ( p fl y. (0 Now we prove an auxiliary statement. Lemma. The binomial identity ( n ( 0 g( 0 holds for every polynomial g( with degree less than n. Proof: Every polynomial g( of degree t can be represented in the following form t g( c l ( l, ans c l are some coefficients. For every l < n ( n ( ( l 0 l0 ( l (... ( l +, l ( ( n ( l ( n!!!(n!( l! n ( n!!!(n!( l! l l n! ( (n l! (n l! (n!( l! l n! ( n l ( 0, ( (n l! n l

7 therefore, ( n ( g( 0. 0 Finally, using the evident asymptotic ( 5 p fl 4(2q O, as q, q along with Lemma and the equality (0, we obtain the statement of Theorem 3.