Average Coset Weight Distributions of Gallager s LDPC Code Ensemble

1 Average Coset Weight Distributions of Gallager s LDPC Code Ensemble Tadashi Wadayama Abstract In this correspondence, the average coset eight distributions of Gallager s LDPC code ensemble are investigated. Gallager s lo-density parity-chec (LDPC) code ensemble consists of regular m n-ldpc matrices ith column eight j and ro eight. The average coset eight distribution can be derived by enumerating the number of parity chec matrices in the ensemble satisfying certain conditions. Based on combinatorial arguments, a formula of the average coset eight distribution ill be proved. From the formula, e can sho some properties of the average coset eight distributions such as equivalence classes of syndromes, symmetry of the distributions, and a loer bound on coset eight. Index Terms: coset, ensemble average, LDPC code, eight distribution I. INTRODUCTION The pioneering ors on lo-density parity-chec (LDPC) codes by Gallager is summarized in [1]. The boo includes a number of important ideas and concepts such as the definition of LDPC codes, some methods for evaluating the (ML and iterative) decoding performance of LDPC codes, and also the sum-product algorithm. The average eight distribution of an LDPC code ensemble is one of such significant outcomes. Gallager defined an ensemble of (j, ) regular LDPC codes here j and denote the column and the ro eights of a parity chec matrix, respectively. He also derived an upper bound on the average eight distribution of the ensemble. It has been non that the average eight distribution of a code ensemble is required for ML (or typical set decoding) performance analysis of LDPC codes [], [3]. Recently, Litsyn and Shevelev [4], [5] presented the average eight distribution of several regular/irregular LDPC code ensembles and their asymptotic behaviors. Burshtein and Miller [6] shoed methods for asymptotic enumeration and have derived the average eight distribution of some irregular LDPC ensembles using their technique. The eight distribution of a linear code is a prime parameter of the code hich represents interesting properties such as ML decoding performance, error detection performance, etc.. Hoever, in some applications using a code, e ould lie to no some coset structures (e.g., coset leader [7], covering radius [8]), or the coset eight distributions of a code. Although the coset eight distribution of a specific code is very difficult to derive in general (e.g., even for the first order T.Wadayama is ith Department of Computer Science, Nagoya Institute of Technology, Nagoya, 466-8555, Japan. (e-mail:adayama@nitech.ac.jp). The or as presented in part at International Symposium on Information Theory and Its Applications (ISITA) 004. Reed Muller code), ensemble arguments enable us to obtain the average of a coset eight distribution for some ensembles of codes. Such average distributions give us useful insights on an instance of the ensemble. In this correspondence, e investigate the average coset eight distribution of Gallager s LDPC code ensemble. A coset of an LDPC code is the set of binary n tuples having the same syndrome. The average coset eight distribution can be derived by enumerating the number of parity chec matrices in the ensemble satisfying certain conditions. Based on combinatorial arguments, a formula of the average coset eight distribution ill be proved. From the formula, e can sho some properties of the average coset eight distributions such as equivalence classes of syndromes, symmetry of the distributions, and a loer bound on coset eight. The organization of this correspondence is as follos. Section II presents the definitions of the average coset eight distributions and the Gallager ensemble. Section III shos a derivation of a formula of the average coset eight distribution. Section IV includes some basic properties of the distributions. Section V gives an upper bound on the average distributions. II. PRELIMINARIES A. Average coset eight distributions Let n and m be positive integers such that n > m. For a given binary m n parity chec matrix H and a vector s F m (F is the Galois field ith to elements), the set C(H, s) is defined by {x F n : Hx t s}, hich is called the coset corresponding to s (called a syndrome) 1. Thus, the code defined by H is denoted by C(H, 0) hose size is larger than or equal to n m. The definition of the coset is slightly different from the traditional one because the ran of H is not necessarily m. The number of vectors of eight in the set C(H, s) is denoted by A (H, s): A (H, s) #{x Z (n,) : Hx t s}, [0, n], (1) here Z (n,) denotes the set {z F n : z }. In this correspondence, the size of a set A is represented by #A and the Hamming eight of a vector x is denoted by x. The notation [a, b] denotes the set of consecutive integers from a to b. The set of numbers {A (H, s)} n 0 is referred to as the coset eight distribution of C(H, s). 1 In this correspondence, a vector is a ro vector and denoted by a boldface letter. The notation x t means the transpose of x (i.e., x t is a column vector).

We then consider the average of A (H, s) over an ensemble of parity chec matrices. For a given ensemble G, the average coset eight distribution {Ã(s)} n 0 is given by à (s) H G P (H)A (H, s) H G P (H)#{x Z (n,) : Hx t s} H G P (H) z Z (n,) I[Hz t s], () here I[condition] gives 1 if condition is true; otherise it gives 0. The probability P (H) is assumed to be 1/(#G). By changing the order of summation and using P (H) 1/(#G), e have à (s) P (H)I[Hz t s] z Z (n,) H G #{H G : Hzt s} #G z Z (n,). (3) Thus, evaluation of Ã(s) is equivalent to enumeration of the number of parity chec matrices in the set {H G : Hz t s}. B. Gallager ensemble Let j and be positive integers such that n/ m/j and n/ is an integer. The Gallager ensemble is an ensemble of parity chec matrices ith column eight j and ro eight, respectively. Let H be the binary (m/j) n matrix satisfying { h 1, if r c rc (4) 0, otherise, here the element in H at the r th ro and the c th column is denoted by h rc (r [0, m/j 1], c [0, n 1]). Note that the column and ro indices start from 0. In other ords, H has the folloing form 11 1 11 1 H. 11 1, (5) here m/j copies of consecutive ones are placed on the main diagonal of H and the remaining elements are set to zero. Let us define G by a set of binary m n matrices such that G H : H σ 1 (H ) σ (H ). σ j (H ), σ i π n (i [1, j]), (6) here σ i π n (i [1, j]) represents a column permutation. The symbol π n denotes the set of all the permutations on n elements, hich has size n!. Thus, the size of G is (n!) j. We no assign equal probability P (H) 1/(n!) j for each element in G. The set G ith this probability assignment is called the Gallager ensemble [1]. Note that the ensemble defined above is not exactly the same as Gallager s original ensemble because the permutation σ 1 is fixed to be the identity permutation in Gallager s original definition. Since the ensemble G can be obtained by applying a permutation to the original ensemble, the ensemble G can be regarded as a natural extended ensemble of the original ensemble. III. DERIVATION OF AVERAGE COSET WEIGHT DISTRIBUTIONS Assume that a syndrome s F m is partitioned to j sub-syndromes such that s (s 1, s,..., s j ) here s i F (m/j) (i [1, j]). We also assume that z is a vector of length n ith eight. We no count the number of permutations hich satisfy σ i (H )z t s i. (7) This number can be regarded as the number of (m/j) n sub-matrices satisfying the given parity chec constraints corresponding to s i. It is easy to see that the above equality is equivalent to H σ 1 i (z ) t s i, (8) here σ 1 i is the inverse permutation of σ i. The above equivalence is obtained by applying σ 1 i (from the left) to the identity H (z ) t σ i (H )σ i ((z ) t ). Due to the one to one correspondence beteen σ i and σ 1 i, the number of permutations satisfying (7) is exactly the same as the number of permutations satisfying (8). Suppose that v Z (n,) satisfies H v t s i. The number of permutations hich satisfy σ 1 i (z ) v is given by #{σ 1 i π n : σ 1 i (z ) v}!(n )!. (9) The reason is as follos: The ones in z should be mapped to ones in v by σ 1 i. The number of possible ays to permute ones is!. In a similar ay, the number of possible ays such that the zeros in z are mapped to zeros in v is (n )!. Thus, the total number of permutations hich satisfy σ 1 i (z ) v is given by!(n )!. The next lemma summarizes the above discussion. Lemma 1: The average coset eight distribution {Ã(s)} n 0 is given by for any s F m à (s) ( ) j n F (s i, ) (10) i1 here F (s i, ) is defined by F (s i, ) 1 ( n )#{v Z (n,) : H v t s i } (11) for s i F m/j and [0, n]. (Proof) Using the equality #{σ i π n : σ i (H )z t s i }!(n )!#{v Z (n,) : H v t s i }, (1)

3 e have F (s i, ) 1 ( n )#{v Z (n,) : H v t s i } (13)!(n )! #{v Z (n,) : H v t s i }(14) n! 1 n! #{σ i π n : σ i (H )z t s i }. (15) Using the fact that the number of permutations (1) is independent from the choice of z Z (n,), e obtain à (s) #{H G : Hzt s} (16) #G z Z (n,) j 1 n! #{σ i π n : σ i (H )z t s i }(17) z Z (n,) i1 ( n ) j F (s i, ). (18) i1 We can evaluate F (s i, ) using the eight enumerators of even and odd eight vectors of length. The next lemma explains the details. Lemma : For any s i F m/j and [0, n], the equality F (s i, ) 1 ( n ) [ α (m/j) s i (x) ], (19) holds, here the notation [f(x)] denotes the coefficient of f(x) corresponding to x. The polynomials α (x) and β (x) are defined by α (x) (1 + x) + (1 x) (0) β (x) (1 + x) (1 x). (1) (Proof) The polynomial α (x) is the eight enumerator of the even eight vectors of length, namely α (x) u F u is even x u () and β (x) is the eight enumerator of the odd eight vectors: Let us denote s i by β (x) u F u is odd x u. (3) s i (s (i) 1, s(i),..., s(i) m/j ), (4) here s (i) l F (l [1, m/j]). Suppose that a vector v F n satisfies H v t s i and it is denoted by v (v 1, v,..., v m/j ), (5) here v l F (l [1, m/j]). Due to the form of H, it is evident that v l is any even eight vector of length if s (i) l 0; otherise (i.e., if s (i) l 1), v l is any odd eight vector of length. Therefore, the eight enumerator of v satisfying H v t s i is given by v F n H v t s i x v α (m/j) si (x). (6) From the definition of F (s i, ) and this eight enumerator (6), e obtain the claim of the lemma. Combining the above to lemmas, e finally obtain the folloing theorem. Theorem 1 (Average coset eight distribution): The average coset eight distribution {Ã(s)} n 0 of the coset corresponding to s F m is given by ( ) j n 1 [ ( n ) à (s) α (m/j) si (x). (7) ] i1 Remar 1: The derivation of the average eight distribution à (0) has already been discussed in Gallager s boo [1]. The main difference beteen Gallager s result and Theorem 1 is the introduction of the odd eight enumerator β (x) hich is required for the case here a syndrome s is not the zero vector. IV. SOME PROPERTIES OF AVERAGE COSET WEIGHT A. Equivalence class DISTRIBUTION Consider to syndromes s (s 1,..., s j ) F m and s (s 1,..., s j ) F m. We say that s and s are equivalent if there exists a permutation σ π j satisfying σ( s 1,..., s j ) ( s 1,..., s j ). (8) From Theorem 1, it is evident that à (s) Ã(s ) holds if s and s are equivalent. This equivalence relation naturally introduces equivalence classes of syndromes. Let {S i } ζ i1 be equivalence classes such that ζ F m S i, (9) i1 S a S b for any a, b(a b) [1, ζ] (30) and any to syndromes in S i are equivalent. The number of distinct equivalence classes ζ can be obtained by enumerating class representatives in the folloing ay. Lemma 3: The number of distinct equivalence classes ζ is given by j ζ #{(b 1, b,..., b j ) : b l [0, j], b l m, b 1 b j }. (31) (Proof) Consider an equivalence class S i and any s (s 1,..., s j ) S i. For each S i, there exists a unique j tuple (b 1,..., b j ) such that there is a permutation σ π j satisfying l1 σ( s 1,..., s j ) (b 1,..., b j ), (3) here b 1 b j. Note that (b 1,..., b j ) must satisfy j l1 b l m. Such a j tuple is called a class representative. One to one correspondence beteen the class representatives and the equivalence classes yields the claim of the lemma.

4 Example 1: Table I presents the average coset eight distribution of the Gallager ensemble ith parameters n 18, m 6, j, 6. The first and second columns represent the class representatives and the average coset eight distribution {Ã(s)} n 0 for each equivalence class, respectively. In this case, there are 10 equivalence classes. B. Zeros in distributions As e have seen in Table I, some equivalence classes may have the all zero distribution. This means that C(H, s) for a syndrome s in such a class. Moreover, e can observe that, for a given s, à (s) 0 for even or for odd. The next lemma justifies these observations. Lemma 4: The folloing statements characterize the zeros in the average coset eight distributions. (i) If s i is even for all i [1, j], then the folloing statements hold: à (s) 0 holds for odd [0, n]. à (s) 0 holds for even [0, n]. (ii) If s i is odd for all i [1, j], then the folloing statements hold: à (s) 0 holds for even [0, n]. à (s) 0 holds for odd [0, n]. (iii) Otherise, à (s) 0 holds for all [0, n]. (Proof) If a polynomial f(x) satisfies [f(x)] 0 for even (resp. odd) and [f(x)] 0 for odd (even), f(x) is said to be even (odd). Assume that s i is even for all i [1, j]. The polynomial α a(x) is even for any positive integer a and βb (x) is even for any positive even integer b. Since the product of even polynomials is also even, α (m/j) s i (x) is even for i [1, j]. This fact and Theorem 1 lead to claim (i). We then suppose that s i is odd for all i [1, j]. Since β b(x) is odd for any positive odd integer b and the product of even and odd polynomials is odd, α (m/j) si (x) is odd for i [1, j]. This implies claim (ii). Finally, e consider the case of (iii). Let p and q be positive integers such that s p is even and s q is odd. From the above arguments, e have [α (m/j) s p (x)β s p (x)] [α (m/j) s q (x)β s q (x)] 0 (33) for all [0, n]. Claim (iii) follos from the equality. Remar : Lemma 4 shos that some cosets must be empty. For example, it is a direct consequence of Lemma 4 that C(H, s) is an empty set if s 1 or m 1 for any H G. This implies that any parity chec matrix in G has ran strictly smaller than m. C. Symmetry of distributions A coset eight distribution is said to be symmetric if à (s) Ãn (s) holds for any [0, n]. When is even, e can prove that any average coset eight distribution is symmetric. Lemma 5: If is even, any distribution is symmetric. Namely, à (s) Ãn (s), [0, n] (34) holds for any s F m. (Proof) Let f(x) be a polynomial of degree δ f. If [f(x)] [f(x)] δf holds for [0, δ f ], f(x) is said to be symmetric. It is easy to prove that the product of to symmetric polynomials is also symmetric. When is even, both of α (x) and β (x) are symmetric. Thus, the polynomial α (m/j) s i (x) is also a symmetric polynomial of degree n. This means that [α (m/j) s i (x)] [α (m/j) s i (x)] n (35) holds for i [1, j]. Combining this equality and Theorem 1, e immediately have the claim of the lemma. When is odd, most coset eight distributions are not symmetric. This is because both of α (x) and β (x) are not symmetric hen is odd. Hoever, some cosets have symmetric distributions. The next lemma explains the symmetric property of the distributions for odd. Lemma 6: When is odd, à (s) Ãn (s ), [0, n] (36) holds if s (s 1,..., s j ) is equivalent to s + 1(m). The notation 1 (a) denotes the all-one vector (1, 1,..., 1) F a of length a. (Proof) Since equivalent syndromes have the same distribution, it is enough to consider the case here s s + 1 (m) ithout loss of generality. Assume that x C(H, s). In this case, e have H(x + 1 (n) ) t Hx t + H1 (n)t (37) s + 1 (m) (38) s (39) because H1 (n)t 1 (m) for odd. This equality implies and thus {x + 1 (n) : x C(H, s)} {x : x C(H, s )} (40) A (H, s) A n (H, s ), [0, n] (41) for any H G and s F m. Averaging (41) over ensemble G, e obtain the claim of the lemma. By letting s s, the lemma gives the condition here a distribution is symmetric. In other ords, if the binary complement of s (i.e., s + 1 (m) ) is included in an equivalence class hich contains s, then Ã(s) is symmetric. Example : Table II presents the average coset eight distributions of the Gallager ensemble ith parameters n 14, m 4, j, 7 (i.e., a case of odd ). We can see that the equivalence classes ith class representatives (0, 0), (, ) have asymmetric distributions and the classes (, 0), (1, 1) have symmetric distribution. This is because the binary complement of a syndrome in the class (, 0) or (1, 1) becomes a member of the same equivalence class. The pair of syndromes s in class (0, 0) and s in class (, ) satisfies à (s) Ãn (s ).

5 TABLE I AVERAGE COSET WEIGHT DISTRIBUTION OF THE GALLAGER ENSEMBLE WITH PARAMETERS: n 18, m 6, j, 6 Class Average coset eight distribution (0, 0) {1, 0, 5 880 18683 598950 598950 18683 880 5, 0,, 0,, 0,, 0,, 0,, 0,, 0,, 0, 1} 17 17 1547 1 1 1547 17 17 (1, 0) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (, 0) {0, 0, 180 310 181718 60580 60580 181718 310 180, 0,, 0,, 0,, 0,, 0,, 0,, 0,, 0, 0} 17 17 1547 1 1 1547 17 17 (1, 1) {0,, 0, 500 63368 443556 665500 443556 63368 500, 0,, 0,, 0,, 0,, 0,, 0,, 0,, 0} 51 119 1 1 1 119 51 (3, 0) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (, 1) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (3, 1) {0, 0, 0, 900 64080 435564 677600 435564 64080 900, 0,, 0,, 0,, 0,, 0,, 0,, 0, 0, 0} 17 119 1 1 1 119 17 (, ) {0, 0, 144 3380 5317636 6063 6063 5317636 3380 144, 0,, 0,, 0,, 0,, 0,, 0,, 0,, 0, 0} 17 17 4641 1 1 4641 17 17 (3, ) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (3, 3) {0, 0, 0, 97 64800 47716 68990 47716 64800 97, 0,, 0,, 0,, 0,, 0,, 0,, 0, 0, 0} 17 119 1 1 1 119 17 TABLE II AVERAGE COSET WEIGHT DISTRIBUTION OF THE GALLAGER ENSEMBLE WITH PARAMETERS: n 14, m 4, j, 7 Class Average coset eight distribution (0, 0) {1, 0, 5 37303 314608 3963 34300, 0,, 0,, 0,, 0, 13 143 49 49 143 (1, 0) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (, 0) {0, 0, 94 35770 308 308 35770, 0,, 0,, 0,, 0, 13 143 49 49 143 (1, 1) {0, 7 1001 1001, 0, 91, 0,, 0, 858, 0,, 0, 91, 0, 7, 0} (, 1) {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (, ) {0, 0, 343 34300 3963 314608 37303, 0,, 0,, 0,, 0, 13 143 49 49 143 343, 0,, 0, 0} 13 94, 0,, 0, 0} 13 5, 0,, 0, 1} 13 D. A loer bound on coset eight In some cases, it is enough to no the coset eight (i.e., eight of the smallest eight vector in a coset) instead of hole eight distributions. We here discuss a loer bound of the coset eight hich depends only on the eight of a syndrome. Let U(H, s) be the coset eight hich is defined by { U(H, s) minx C(H,s) x if C(H, s) (4) if C(H, s), for H G and s F m. The next theorem shos that U(H, s) can be loer bounded by the maximum of s i. This bound is easy to compute ithout evaluation of the distribution formula (7) given by Theorem 1. Theorem (Syndrome eight bound): The coset eight U(H, s) is loer bounded by max s i U(H, s) (43) 1 i j for any s F m and H G. (Proof) We ill prove the folloing to claims. The combination of these gives the claim of the theorem. (i) (ii) The coset eight U(H, s) is loer bounded by L(s) U(H, s) for any s F m and H G here { L(s) n + 1, if à (s) 0 for [0, n], min{ [0, n] : Ã(s) 0}, otherise. (44) The folloing inequality holds for any s F m. max s i L(s) (45) 1 i j We first consider claim (i). Suppose that there exist Ĥ G and u C(Ĥ, s) hose eight is smaller than L(s). It contradicts the assumption à u (s) 0. Thus claim (i) has been proved. We then discuss claim (ii). For to nonzero polynomials f(x) and g(x), the equality ldeg(f(x)g(x)) ldeg(f(x)) + ldeg(g(x)) (46) holds, here the notation ldeg(p(x)) denotes the loest degree of the nonzero terms of a polynomial p(x). By using this equality and the initial conditions e can prove ldeg(α (x)) 0 (47) ldeg(β (x)) 1, (48) ldeg(α (m/j) si (x)) s i. (49) It is obvious that (49) implies that F (s i, ) 0 if < s i. Thus, e can see that j F (s i, ) 0 (50) i1 holds if < max 1 i j s i. It follos that à (s) 0 if < max 1 i j s i, hich implies claim (ii). The maximum of the coset eights except for empty cosets, ρ(h), is called the covering radius [8], hich is defined by ρ(h) s F m max U(H, s). (51) :C(H,s) By letting s 1 (m) and using Theorem, e immediately have the folloing loer bound on the covering radius: m/j ρ(h). (5)

6 E. Existence proof using average distributions The average coset eight distribution of G can be used for proving existence of a parity chec matrix hich satisfies certain conditions. 1) Moment function: Let g() be a real valued function defined over [0, n]. A moment function of g() is given by E[H, s, g] A (H, s)g(). (53) 0 We no consider the ensemble average of the moment function Ẽ[s, g] H G P (H)E[H, s, g] (54) H G P (H) A (H, s)g() (55) 0 P (H)A (H, s) (56) g() 0 H G à (s)g(). (57) 0 From the standard arguments of the ensemble average, there exist parity chec matrices H, H G hich satisfy E[H, s, g] E[H, s, g] Ẽ[s, g] (58) Ẽ[s, g]. (59) This argument is the basis of the existence proof using average distributions. Example 3: Suppose that a binary n tuple x is generated from a binary i.i.d. source ith probability P r[x i 1] p. For a given H G, the probability of the event that x has the syndrome s is given by P r[hx t s] A (H, s)p (1 p) n. (60) 0 Letting g() p (1 p) n, e have the ensemble average of P r[hx t s] in such a ay: H G P (H)P r[hx t s] Ẽ[s, g] (61) à (s)p (1 p) n.(6) 0 ) Accumulated coset eight: Let B τ (H, s) be the accumulated coset eight distribution defined by B τ (H, s) τ A (H, s) (63) 0 for τ [0, n], s F m, H G. Averaging B τ (H, s) over the ensemble G, e obtain the average accumulated coset eight distributions in the folloing ay: B τ (s) H G P (H)B τ (H, s) (64) H G P (H) τ A (H, s) (65) 0 τ P (H)A (H, s) (66) 0 H G τ à (s). (67) 0 When B τ (s) < 1, it is obvious that there exists a parity chec matrix H G hich gives B τ (H, s) 0 since at least one matrix in G must give the value belo the ensemble average. The next lemma is based on this idea. Lemma 7: There exists a parity chec matrix H G hich satisfies here T (s) T (s) < U(H, s), (68) { 0, if à (s) 0, ( [0, n]) max{τ [0, n] : B τ (s) < 1}, otherise. (69) (Proof) Suppose that τ is any integer satisfying B τ (s) < 1. From this assumption and the definition of Bτ (s), e can sho that there exists H G satisfying B τ (H, s) 0. This implies τ < U(H, s). Letting τ T (s), e have the claim of the lemma. Remar 3: The typical minimum distance of the Gallager ensemble [1] can be obtained by finding the maximum τ satisfying B τ (0) <. This is because the all zero vector is alays included in C(H, 0) for any H G and the zero vector must be excluded from the candidates of the minimum eight vectors. V. AN UPPER BOUND ON AVERAGE COSET WEIGHT DISTRIBUTION Theorem 1 yields a method to evaluate the exact average coset eight distributions. Hoever, for large n (especially for the asymptotic case), expansion of the polynomial α (m/j) si (x) is computationally difficult in general. We here employ Gallager s bounding method to avoid the expansion of the polynomial. Plugging x t into α (m/j) s i (x), e have α (m/j) s i ( t )β s i (t ) t Q(), (70) 0 here t is a real number. The symbol Q() denotes [α (m/j) s i (x)]. For any t R and [0, n], the inequality t Q() t Q() (71) 0

7 holds, here R denotes the set of real numbers. From this inequality, an upper bound on Q() Q() α (m/j) s i ( t )β s i (t )/ t (7) ν (t, s i ) t (73) is obtained for any t R here ν (t, u) (m/j u) log α ( t ) + u log β ( t ). (74) The next lemma gives an upper bound on Ã(s). Lemma 8: For any s F m, j à (s) [ (j 1)H( n n )+ 1 n (75) holds here H(p) is the binary entropy function defined by i1 inf t R{ν (t, s i ) t}+ɛ(n) H(p) p log p (1 p) log (1 p) (76) and ɛ(n) (j 1) log (n + 1). (77) n (Proof) From (73), e immediately have [ α (m/j) s i ] (x) inf t R{ν (t, s i ) t}. (78) Using inequality (78), à (s) can be upper bounded in the folloing ay: ( ) j n 1 [ à (s) ( n α i1 ) (m/j) si ] (x) (79) ( ) (j 1) j n inf t R{ν (t, s i ) t} (80) i1 ( ) (j 1) n j inf t R{ν (t, s i ) t} i1. (81) Applying a loer bound on the binomial coefficient [9] ( ) n 1 (n + 1) nh( n ) (8) to (81), e obtain the claim of the lemma. Remar 4: Applying the asymptotic enumeration methods by Burshtein and Miller [6] to Theorem 1, e can obtain the same upper bound. Furthermore, by using the method in [6], e can sho that the upper bound is tight in the sense that the logarithm of both sides of (75) normalized by n approach the same value. VI. CONCLUSION In this correspondence, a formula of average coset eight distributions of Gallager ensemble has been derived. It is shon that some properties related to coset eight can be proved based on the average coset eight distributions. There are various types of regular/irregular LDPC matrix ensembles (see [4], [5]). The derivation of average coset eight distributions for such ensembles is an interesting problem. Some are easy to derive. (As an example, in Appendix, derivation of the distributions of Ensemble E [4] is given.) Hoever, many others are still open because the method developed in this correspondence may not be extended to other ensembles. ] APPENDIX Consider the case of Ensemble E ([4], p. 903 )), hich is referred to as E in this correspondence. The ensemble E is the ensemble of m n binary matrix ith ro eight. From (3), the average coset eight distribution of E is given by à (s) #{H E : Hzt s} ( n m. (83) z Z ) (n,) In order to evaluate (83), e need to count the number of matrices in the set {H E : Hz t s}. Assume that a binary n tuple z has eight. From a simple combinatorial argument, it is easy to derive that D 0 #{h Z (n,) : hz t is even} / ( )( ) n i i i0 (84) D 1 #{h Z (n,) : hz t is odd} ( ) n D 0. (85) Let s (s 1, s,..., s m ) F m. Since the size of {H E : Hz t s} is equal to D s1 D s D sm, the number of matrices #{H E : Hz t s} is given by ( ) m n [γ(n,, )] m s [1 γ(n,, )] s, (86) here γ(n,, ) is defined by γ(n,, ) 1 ( n ) / i0 ( i )( ) n i (87) (see also discussion on the average eight distribution of Ensemble E ([4], p.903 )). Note that the value of (86) depends only on the eight of z. Thus, combining the above results, e have the average coset eight distribution of E: ( ) à (s) [γ(n,, )] m s [1 γ(n,, )] s n. (88) Acnoledgment The author ould lie to than Dr. Jun Muramatsu of NTT Communication Science Laboratories for inspiring discussion on LDPC code ensembles. The author also ishes to than the anonymous revieers for valuable comments. REFERENCES [1] R.G.Gallager, Lo Density Parity Chec Codes. Cambridge, MA:MIT Press 1963. [] D.J.C. MacKay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory, vol.45, pp.399 431, March 1999. [3] G.Miller and D.Burshtein, Bounds on maximum-lielihood decoding error probability of lo-density parity-chec codes, IEEE Trans. Inform. Theory, vol.47, pp.696 710, Nov. 001. [4] S.Litsyn and V. Shevelev, On ensembles of lo-density parity-chec codes: asymptotic distance distributions, IEEE Trans. Inform. Theory, vol.48, pp.887 908, Apr. 00. [5] S.Litsyn and V. Shevelev, Distance distributions in ensembles of irregular lo-density parity-chec codes, IEEE Trans. Inform. Theory, vol.49, pp.3140 3159, Nov. 003.

8 [6] D.Burshtein and G. Miller, Asymptotic enumeration methods for analyzing LDPC codes, IEEE Trans. Inform. Theory, vol.50, pp.1115 1131, June 003. [7] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland 1977. [8] G. Cohen, I. Honala, S. Litsyn and A. Lobstein, Covering Codes. Amsterdam, The Netherlands: North-Holland 1997. [9] G. Duec and J. Körner, Reliability function of a discrete memoryless channel at rates above capacity, IEEE Trans. Inform. Theory, vol.it-5, pp.8 85, Jan. 1979 Biography Tadashi Wadayama as born in Kyoto,Japan,on May 9,1968. He received the B.E., the M.E., and the D.E. degrees from Kyoto Institute of Technology in 1991, 1993 and 1997, respectively. Since 1995, he has been ith Oayama Prefectural University as a research associate. Since 004, he has been ith Nagoya Institute of Technology as an associate professor. His research interests are in coding theory, information theory, and digital communication/storage systems. He is a member of IEICE, SITA and IEEE.