Bounds on the Error Probability of Block Codes over the ... - IEEE Xplore

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Bounds on the Error Probability of Block Codes over the q -Ary Erasure Channel Gianluigi Liva, Member, IEEE, Enrico Paolini, Member, IEEE, and Marco Chiani, Fellow, IEEE

Abstract—In this paper, tight bounds on the block error probability of linear block codes over order-q finite fields for the q-ary erasure channel, under maximum-likelihood (ML) decoding, are developed. Upper bounds are obtained for uniform parity-check ensembles, sparse parity-check ensembles, general parity-check ensembles (e.g., Gallager regular nonbinary lowdensity parity-check ensembles), and for any given linear code with known distance spectrum. The tightness of the upper bounds is confirmed both by the comparison with simple lower bounds and, for Gallager low-density parity-check ensembles, by extensive Monte Carlo simulations. Exploiting the derived bounds, it is shown how already for short blocks and small q > 2 sparse ensembles attain block error probabilities close to those of idealized maximum distance separable (MDS) codes, down to low error probabilities, whereas in the same regime binary codes show visible losses with respect to the Singleton bound. Thanks to the accurate performance estimates, the developed bounds can support the design of near-optimum erasure correcting codes with short and moderate lengths. Index Terms—Block error probability, finite fields, low-density parity-check (LDPC) codes, maximum-likelihood decoding, q-ary erasure channel, Singleton bound, union bound.

I. I NTRODUCTION

T

HE binary erasure channel (BEC) [1] is often used as a simplified model for data losses in wireless/wired digital networks [2]–[4]. Thus, the design of powerful coding schemes under low-complexity decoding for the BEC has been a rich research area in the past years [2], [3], [5]–[7]. In this framework, a class of codes that gained a particular attention is the one of low-density parity-check (LDPC) codes [8], which approach the erasure channel capacity under belief propagation (BP) decoding for very large block lengths [2], [5], [9], [10]. In [11]–[18], low-complexity maximum-likelihood (ML) erasure decoders for LDPC codes were proposed that allow achieving, in combination with a judicious code design, performances close to those of idealized maximum distance separable (MDS) codes.

Manuscript received July 12, 2012; revised November 3, 2012. The associate editor coordinating the review of this paper and approving it for publication was A. Graell i Amat. G. Liva is with the Institute of Communication and Navigation of the Deutsches Zentrum fur Luft- und Raumfahrt (DLR), 82234 Wessling, Germany (e-mail: [email protected]). E. Paolini and M. Chiani are with CNIT, DEI, University of Bologna, 47521 Cesena (FC), Italy (e-mail: {e.paolini, marco.chiani}@unibo.it). The research leading to these results has received funding in part from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n. 288502 and in part from the Deutscher Akademischer Austausch Dienst (DAAD) under DLR-DAAD fellowship No. 83. Digital Object Identifier 10.1109/TCOMM.2013.032013.120504

Bounds on the performance (in terms, e.g., of block error probability) of random and sparse graph-based ensembles over erasure channels have been the subject of past and recent research. For example, the block error probability over the BEC for random (linear) binary ensembles was analyzed in [1], [19], [20]. In [21] the performance limits over the block erasure channel (BLEC) have been investigated. The error probability of LDPC code ensembles in the finite length regime under both BP and ML decoding was analyzed in [22] over the BEC. Moreover, tight upper and lower bounds on the error probability for some classes of Luby-transform (LT) codes were established in [23]. Upper bounds on the block error probability of nonbinary linear codes over memoryless channels, under ML decoding, were also developed in [24] and [25]. Specifically, in [24] performance bounds under ML decoding for three coset ensembles of block codes based on underlying expurgated LDPC code ensembles were derived, assuming transmission over a discrete memoryless (in general not symmetric) channel. The considered expurgated ensembles of underlying LDPC codes were binary, q-ary in the sense of Gallager, and based on finite fields. The coset LDPC code ensembles based on finite fields over discrete memoryless channels were later analyzed in [26] under iterative decoding, via density evolution and extrinsic information transfer (EXIT) charts. In [25], upper bounds on the block error probability of nonbinary linear block code ensembles over memoryless symmetric channels were obtained and applied to Gallager’s q-ary regular LDPC codes. The definition of channel symmetry proposed in [25] represents a generalization of the standard definition of symmetry for memoryless binary-input output-symmetric (MBIOS) channels, and ensures independence of the conditional error probability on the transmitted codeword. The memoryless q-ary erasure channel (q-EC) represents an extension of the BEC. In a q-EC, the channel input alphabet X has cardinality |X | = q and each symbol is either correctly received or erased. The q-EC may serve as a reference model for data losses in digital networks, as its binary counterpart [27]–[29]. It has also been adopted to model optical communication links under q-ary pulse position modulation (PPM) and photon counting receivers in absence of background radiation [30], [31], [32, Chapter 4]. Furthermore, it gives an alternative representation of a binary BLEC [21] where blocks of log2 q bits are either correctly received or erased. In [33] the error exponents of random and MDS codes over the q-EC were derived and compared.

c 2013 IEEE 0090-6778/13$31.00 

LIVA et al.: BOUNDS ON THE ERROR PROBABILITY OF BLOCK CODES OVER THE Q-ARY ERASURE CHANNEL

In this paper, we develop tight bounds on the block error probability of linear block codes over the q-EC under ML decoding, assuming that the field order of the code matches the channel input alphabet size q. We derive upper and lower bounds for dense (i.e., uniform) parity-check ensembles and sparse parity-check ensembles, and upper bounds both for codes whose distance spectrum is known and for general parity-check ensembles with known average distance spectrum. In this latter case, the tightness of the bounds for Gallager LDPC ensembles is confirmed by extensive Monte Carlo simulations. Exploiting the developed bounds, we show that sparse ensembles attain block error probabilities close to those of idealized MDS codes, also for short blocks and relatively small q > 2, down to low error probabilities, whereas binary codes suffer for visible losses with respect to the Singleton bound. The developed bounds can thus be used to support the design of near-optimum finite length codes, accurately predicting their performance over the q-EC. This paper is organized as follows. Section II introduces the notation and reviews some preliminary results. Tight upper and lower bounds on the average block error probability of sparse and dense parity-check ensembles are derived in Section III. A tight union upper bound on the block error probability of a generic linear block code ensemble is provided in Section IV. Conclusions follow in Section V. II. P RELIMINARIES A. Definitions The memory-less BEC is characterized by a binary input alphabet X = {0, 1} and a ternary output alphabet Y = {0, 1, E}. The channel transition probabilities are pY |X (0|0) = pY |X (1|1) = 1 −  and pY |X (E|0) = pY |X (E|1) = , where E represents an erasure and  is the channel erasure probability. In a q-EC, the input alphabet X has cardinality q. We assume that q = 2l and that the input alphabet symbols are the elements of Fq , the finite field of order q. Thus, X = {0, 1, α1 , . . . αq−2 } where α is a primitive element of Fq . The output alphabet is Y = {0, 1, α1 , . . . αq−2 , E}, and its cardinality is q + 1. The channel transition probabilities are pY |X (x|x) = 1 −  and pY |X (E|x) = , ∀x ∈ X (Figure 1). In the following we denote by wH (v) the Hamming weight of a vector v with elements in Fq(i.e., the number of its nonzero elements) and by v, w = i vi wi the inner product between v and w, where all operations are in Fq . Moreover, we denote by Fnq the n-dimensional vector space of n-tuples over Fq (with standard addition and scalar multiplication). The finite field order of the linear block code G used to communicate over the q-EC is assumed to match the channel input alphabet size. We define the support of a codeword x = (x0 , x1 , . . . , xn−1 ) ∈ G as the set of its nonzero coordinates. Formally, supp(x) = {j ∈ {0, 1, . . . , n − 1} s.t. xj = 0}. We also define the support set of x ∈ G as Sx = {x ∈ G s.t. supp(x ) = supp(x)}. This is the set of codewords in G having the same support as x. (Note that x ∈ Sx by default.) Moreover, we define the weight of a support as the Hamming weight of the corresponding codewords. When a codeword of G is transmitted over the q-EC, the resulting erasure pattern IE = {i1 , i2 , . . . , ie } ⊆ {0, 1, . . . , n −

0

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 Æ  Æ  Æ

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 9 9 )  

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^ ^ R z

Fig. 1.

 Æ  Æ

αq−2

E

The q-ary erasure channel model.

1}, of generic size e, identifies an m × e sub-matrix of the parity-check matrix, composed of the e columns of the paritycheck matrix with indexes in IE . ML decoding fails if and only if the rank of this m × e sub-matrix is less than e or, equivalently, supp(x) ⊆ IE for some nonzero codeword x. The corresponding block error probability over a q-EC with erasure probability  is denoted by PB (G, ). We further denote by π(G, e) the average decoding failure probability given that there are exactly e erased symbols in the received word and assuming as equiprobable all erasure patterns of cardinality e. Throughout the paper we consider several parity-check ensembles, i.e., ensembles of linear block codes induced by random parity-check matrices. More specifically, the paritycheck ensemble induced by a random matrix H with entries in Fq is composed of the linear block codes over the same finite field representing the null spaces of all possible realizations of H. The probability of each code is the sum of the probabilities of the corresponding realizations of H. Denoting by HIE the m × e sub-matrix of the random matrix H whose columns correspond to the indexes in a size-e erasure pattern IE = {i1 , i2 , . . . , ie }, the probability of a decoding failure is Pr{rank(HIE ) < e}. In particular, we denote by C(n, m, q) the ensemble of linear block codes induced by an m × n parity-check matrix H whose entries are independent and identically distributed (i.i.d.) random variables X ∈ Fq with uniform probability mass function (p.m.f.), i.e., Pr{X = β} = 1/q ∀ β ∈ Fq . We call this ensemble the uniform parity-check ensemble. We also denote by S(n, m, q, p) the ensemble induced by an m × n parity-check matrix H whose entries are i.i.d. random variables X ∈ Fq such that Pr{X = 0} = 1 − p and Pr{X = β} = p/(q − 1) ∀ β ∈ Fq \ {0}. For small p sparse realizations of H are picked with a higher probability than dense ones, hence we call this ensemble the sparse paritycheck ensemble. Moreover, we denote by L(n, J, K, q) the Gallager regular LDPC code ensemble over Fq [8, Ch. 5].

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Here, J represents the weight of each column of H and K the weight of each row of H. Instead of considering a specific parity-check ensemble over Fq , in some cases we will refer to a generic one, in the sense that the parity-check matrix H is characterized by a generic probability distribution. This ensemble will be denoted by G(n, m, q). The previous parity-check ensembles can be seen as particular instances of G(n, m, q). Finally, we define as C (n, k, q) the ensemble of all codes with q k codewords and length n, assuming a uniform probability distribution on such codes. Equivalently, C (n, k, q) is defined by a random code book of q k codewords where each codeword is picked uniformly at random in Fnq . As opposed to any parity-check ensemble G(n, m, q), the ensemble C (n, k, q) also includes nonlinear codes. B. Singleton Bound The Singleton bound [34] is defined as   n n e .  (S) PB (n, m, ) =  (1 − )n−e . e e=m+1

code in C (n, n − m, 2) whose block error probability under ML decoding is upper bounded by the Berlekamp bound. An alternative upper bound on the average block error probability for the ensemble C (n, n−m, 2) is the Gallager random coding bound [35], extended to the ensemble C (n, n − m, q) in [33]: EC (n,k,q) [PB (G, )] < e−nEq, (r)

where Eq, (r) is the random coding exponent for the q-ary codes with rate r = 1 − m/n. On the q-EC, the exponent Eq, (r) is given by ⎧

 ⎪ 0 <  < c ⎨ − ln 1− q +  − r ln q Eq, (r) = D(1 − r, ) c ≤  ≤ 1 − r ⎪ ⎩ 0 >1−r (5) where D(1 − r, ) = (1 − r) ln

(1)

The right-hand side of (1) is the probability that the number e of erased symbols exceeds the number m of equations (assumed linearly independent of each other) and then represents the block error probability of an idealized q-ary linear MDS code with length n and dimension n−m over the q-EC. Hence, it is a lower bound on the block error probability of every linear code of length n and rate r ≥ (n − m)/n. For any parity-check ensemble G(n, m, q), the average block error probability may be then always expressed as

(4)

r 1−r + r ln  1−

is the Kullback-Leibler distance between two Bernoulli distributions with parameters  and 1 − r, and c =

1−r . r(q − 1) + 1

(6)

Both (3) and (4) (with q = 2) represent upper bounds on the block error probability of the best code in C (n, n − m, 2). The bound (4) is known to be less tight than (3) down to low block error probabilities [36]. However, it becomes tighter at very low error probabilities.

(S)

EG(n,m,q) [PB (G, )] = PB (n, m, ) m    n e +  (1 − )n−e EG(n,m,q) [π(G, e)] e e=1

(2)

where EG(n,m,q) [·] denotes expectation over the ensemble. C. Berlekamp and Gallager Random Coding Bounds The performance of binary codes on the BEC was initially analyzed in [1], while the expected performance under ML decoding of a code drawn from C(n, m, 2) was studied in detail in [19], [20]. More specifically, the average block error probability of codes in C(n, m, 2) was upper bounded in [19] as (Berlekamp random coding bound) (S)

EC(n,m,2) [PB (G, )] < PB (n, m, ) m    n e +  (1 − )n−e 2−(m−e) . (3) e e=1 In [22, App. B], the Berlekamp bound is re-derived as a union bound on the expected block error probability of the codes in C(n, m, 2). Since (3) provides an upper bound on the average block error probability for the parity-check ensemble C(n, m, 2), which includes all binary linear block codes of length n and rate r ≥ 1 − m/n, it can be used as an upper bound for the ensemble C (n, n − m, 2) of binary block codes with length n and 2n−m codewords, in the sense that there exists at least one

D. Rank of Matrices with i.i.d. Uniform Elements over Fq For e ≤ m, consider an e × m random matrix A, whose elements are drawn independently at random with uniform distribution in Fq . We define Q = Pr{rank(A) < e} as the probability of A being not full-rank. We also denote by δ = m − e the number of rows in excess w.r.t. the number of columns. The probability Q may be expressed as a function of e, δ, and q and is given by [37], [38]  e  q i−1 1 − e+δ . Q(e, δ, q) = 1 − q i=1

(7)

The following result was developed in [28]. Proposition 1: The probability Q can be bounded above and below as q −δ−1 ≤ Q(e, δ, q)
PB (n, m, ) m    n e +  (1 − )n−e [1 − (1 − (1 − p)m )e ] . e e=1 (10) Proof: We start by proving the upper bound. Consider an erasure pattern IE of size |IE | = e and its corresponding sub-matrix HIE . Via union bound we have ES(n,m,q,p) [π(G, e)] = Pr {rank(HIE ) < e}   = Pr ∃ v ∈ Feq \ {0} s.t. HIE vT = 0T ⎧ ⎫ ⎨ ⎬  (a)   1 ≤ min 1, Pr HIE vT = 0T ⎩ q−1 ⎭ v∈Feq \{0}

e   1  e = min 1, (q − 1)t q − 1 t=1 t    × Pr HIE vT = 0T |wH (v) = t (11) where the 1/(q − 1) factor in (a) is due to the fact that if some v1 ∈ Feq \ {0} exists such that HIE v1T = 0T , then there certainly exist other q − 2 vectors vi ∈ Feq \ {0}, i = 2, . . . , q − 1, such that HIE viT = 0T and such that q−1 T T T i=1 {event that HIE vi = 0 } = {event that HIE v1 = T 0 } (simply multiply v1 bythe  q − 2 elements of Fq \ {0, 1}). Note that there are exactly et (q − 1)t vectors v of length e and Hamming weight t andthat all of them are characterized  by the same probability Pr HIE vT = 0T |wH (v) = t .

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Next, denoting by h the generic row of HIE , due to independence and identical distribution of the entries of HIE we have   Pr HIE vT = 0T |wH (v) = t m = (Pr {h, v = 0|wH (v) = t}) m

= (Pr {X1 + X2 + · · · + Xt = 0})

(12)

where X1 , X2 , . . . , Xt are i.i.d. random variables in Fq each characterized by a p.m.f. pX equal to that of the generic entry of HIE . The expression (12) may be evaluated using the Fourier transform over Abelian groups [39, Chapter 10], as follows. Let Gq be the additive group of Fq and χ be the generic character of Gq , with the convention that χ0 is the principal character of Gq , i.e., χ0 (a) = 1 ∈ C for all a ∈ Gq (as usual, C denotes the field of complex numbers). Due to independence, the p.m.f. pY of Y = X1 + X2 + · · · + Xt is given by the convolution of the p.m.f.s of the t summands. Interpreting p.m.f.s as functions from Gq to C, the Fourier transform pˆY of pY is given by pˆY (χ) = [ˆ pX (χ)]t , where pˆX is the Fourier transform of pX . It is easy to show that1  1 if χ = χ0 pˆ(χ) = pq 1 − q−1 elsewhere from which, applying the inverse Fourier transform, we obtain  t q−1 1 pq Pr{X1 + X2 + · · · + Xt = 0} = + . 1− q q−1 q (13) Substituting back (13) into (12) and then (12) into (11), from (2) we obtain (9). The lower bound may be proved as follows. Considering again an erasure pattern IE of size |IE | = e and its corresponding sub-matrix HIE , the probability ES(n,m,q,p) [π(G, e)] = Pr {rank(HIE ) < e} is lower has atleast bounded by the probability that HIE   one all-zero e column, this latter probability being t=1 et (1 − p)mt (1 − (1 − p)m )e−t = 1 − (1 − (1 − p)m )e . The lower bound (10) is then obtained from (2). Example 1: Figure 2 shows the upper bounds (9) on the average block error probability for S(120, 60, q, p) sparse parity-check ensembles with q = 2, 4, and 16, and matrix densities p = 1/3, 1/4, and 1/5. On the same plot, the lower bounds (10) are provided. Observe that, while in the waterfall region the upper bounds tend to get closer to the Singleton bound for increasing field orders q, in the error floor region the upper bounds become almost independent of q, and are nearly undistinguishable from the corresponding lower bounds (which are independent of the field order). Thus, for sparse parity-check ensembles the average error floor performance is dominated by the performance of the codes in the ensemble having minimum distance 1, i.e., codes having codeword symbols that do not participate in any parity-check equation. The probability of drawing a code with minimum distance 1 drops remarkably by increasing the matrix density p. For instance, for p = 1/3 the error floor appears at block 1 Since we consider fields of order q = 2l , the Fourier transform reduces to the Hadamard transform [40].

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−2

parity-check ensemble C(n, m, q) is derived next. Note that this lower bound does not represent a special case of (10).

10

−3

10

Theorem 2: Let C(n, m, q) be the ensemble of linear block codes induced by an m × n random parity-check matrix H with i.i.d. entries X such that Pr{X = β} = 1/q ∀β ∈ Fq . Then the expected block error probability of a code G picked randomly in C(n, m, q), under ML decoding and over a q-EC with erasure probability , fulfills

−4

Block Error Probability

10

p = 1/5

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p = 1/4

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Singleton Bound F2 ensemble, p = 1/5 - UB F4 ensemble, p = 1/5 - UB F16 ensemble, p = 1/5 - UB Lower Bound, p = 1/5 F2 ensemble, p = 1/4 - UB F4 ensemble, p = 1/4 - UB F16 ensemble, p = 1/4 - UB Lower Bound, p = 1/4 F2 ensemble, p = 1/3 - UB F4 ensemble, p = 1/3 - UB F16 ensemble, p = 1/3 - UB Lower Bound, p = 1/3

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(S)

EC(n,m,q) [PB (G, )] ≥ PB (n, m, ) m    n e +  (1 − )n−e q −(m−e)−1 . e e=1 (15) 0.5

Fig. 2. Upper and lower bounds (9) and (10) on the average block error probability for sparse parity-check ensembles with n = 120 and m = 60, for the binary, 4-ary, and 16-ary erasure channel. Matrix density p set to 1/5, 1/4 and 1/3. Note that the lower bound is independent of the field order.

error probability about 10−9 , while for p = 1/5 the floor rises up at a block error probability about 10−4 . An upper bound on the expected block error probability of a linear block code drawn from the uniform parity-check ensemble C(n, m, q) may be obtained as a corollary of Theorem 1. Corollay 1: Let C(n, m, q) be the ensemble of linear block codes induced by an m × n random parity-check matrix H with i.i.d. entries X such that Pr{X = β} = 1/q ∀β ∈ Fq . Then the expected block error probability of a code G picked randomly in C(n, m, q), under ML decoding and over a q-EC with erasure probability , may be upper bounded as (S)

EC(n,m,q) [PB (G, )] < PB (n, m, ) m   1  n e +  (1 − )n−e q −(m−e) . q − 1 e=1 e (14) Proof: The ensemble C(m, n, q) may be regarded as a special instance of the ensemble S(n, m, q, p) by setting p = 1 − 1/q. Inequality (14) then follows from (9) by observing that for p = 1 − 1/q we have m   t e   1 q − 1 p q 1  e + (q − 1)t 1− q − 1 t=1 t q q−1 q   e 1  e = (q − 1)t q −m q − 1 t=1 t 1 qe − 1 q − 1 qm 1 −(m−e) q < q−1 =

and that

1 −(m−e) q−1 q

< 1 for all q ≥ 2 and e < m.

A lower bound on the expected block error probability decoding of a linear block code drawn from the uniform

Proof: For a general parity-check ensemble G(n, m, q) the expected block error probability is given by (2). In the specific case of the C(n, m, q) ensemble, from Section II-D we have EC(n,m,q) [π(G, e)] = Q(e, m − e, q)  e  q i−1 =1− 1− m . q i=1

(16)

Moreover, Proposition 1 yields q −(m−e)−1 ≤ EC(n,m,q) [π(G, e)]
2, at least down to PB = 10−10 . 2 Since at least one code in C (n, n − m, q) must exist whose performance is upper bounded by the q-ary extension (14) of the Berlekamp bound, at least one code in C (120, 60, q) exists, for any q ≥ 16, approaching very closely the Singleton bound down to low block erasure probabilities. Moreover, due to tightness, extremely good codes (i.e., codes performing very close to the Singleton bound down to low block error probabilities) exist even in the linear (parity-check) ensemble. IV. U NION U PPER B OUND ON THE AVERAGE B LOCK E RROR P ROBABILITY OF A G ENERIC PARITY-C HECK E NSEMBLE In this section, we develop an upper bound on the expected ML performance over the q-EC of a code picked in a generic parity-check ensemble G(n, m, q). The bound represents a tighter version of a bound previously introduced in [22], and its proof follows the main steps of the proof developed in [22]. Consider a binary linear block code G belonging to a generic parity-check ensemble G(n, m,  2), with weight enumerating function (WEF) A(x) = 1 + i≥1 Ai xi where Ai is the multiplicity of codewords with Hamming weight i. Moreover, let dmin be the minimum distance of G. An upper union bound on its block error probability over the BEC under ML decoding is given by [22] (S)

PB (G, ) ≤ PB (n, m, ) 

    m e   n e e Aw n−e n . + min 1,  (1 − ) e w w e=dmin

w=dmin

(18) 2 Note that for the C (120, 60, 2) ensemble we have  = 1/3. Thus, down c to PB ≈ 10−5 the Gallager random coding bound for q = 2 coincides its extension to larger values of q.

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Fig. 4. Lower and upper bounds (15) and (14) on EC(120,60,q) [PB (G, )], for the binary, 4-ary, and 16-ary erasure channel. The Gallager random coding bound (4) with the error exponent (5) for the ensemble C (120, 60, q) is also depicted for the case of large q. Note that the performance under ML decoding of at least one code in C (120, 60, q) for q ≥ 16 is upper bounded by the right-hand side of (14) with q = 16 and is lower bounded by the Singleton bound, thus achieving a near-optimum performance at least down to block error probability 10−10 .

Moreover, denote by A(x) = EG(n,m,2) [A(x)] = 1 +  i i≥1 Ai x the WEF averaged over the G(n, m, 2) ensemble, where Ai is the expected multiplicity of codewords with Hamming weight i. Then, a bound analogous to (18) and holding for the expected block error probability over G(n, m, 2) is given by (S)

EG(n,m,2) [PB (G, )] ≤ PB (n, m, )

 m   e     n e e A w n . +  (1 − )n−e min 1, e w w e=1 w=1

(19)

Notably, the bounds (18) and (19) hold not only for a binary linear block code (ensemble) over the BEC, but also for a nonbinary linear block code (ensemble) constructed on Fq over the q-EC. Next, tighter versions of (18) and (19) are derived for a code (ensemble) on a nonbinary field. Proposition 2: Let G be a linear block code on Fq . For any nonzero codeword x ∈ G, we have |Sx | ≥ q − 1, where |Sx | is the cardinality of Sx . Proof: For any nonzero codeword x ∈ G, let xβ = β x, β ∈ Fq \ {0, 1}. Since supp(xβ ) = supp(x), there exist at least other q − 2 nonzero codewords sharing the same support as x. Theorem 3: Let G be a linear block  code from the ensemble G(n, m, q) and let A(x) = 1 + i≥dmin Ai xi be its WEF, where dmin is the minimum distance of G. Then, the block error probability of G over the q-EC under ML decoding may be upper bounded as

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upper bounded as (S) PB (n, m, )

PB (G, ) ≤ ⎧   m ⎨  1 n e +  (1 − )n−e min 1, ⎩ q−1 e e=dmin

(S)

e  w=dmin

⎫ ⎬

  e Aw n . ⎭ w w (20)

Proof: Let Pr {error} be the probability that an erasure pattern IE punctures the support of a codeword of G, i.e., Pr{error} = Pr{∃x ∈ G s.t. supp(x) ⊆ IE }. Then we can write (S)

PB (G, ) = PB (n, m, ) m  + Pr{|IE | = e} Pr{error| |IE | = e} e=1 (S)

= PB (n, m, ) +

m    n e  (1 − )n−e e e=1

× Pr{error| |IE | = e}   m  n e (S) = PB (n, m, ) +  (1 − )n−e e e=dmin

× Pr{∪x∈G {supp(x) ⊆ IE }| |IE | = e} .

(21)

EG(n,m,q) [PB (G, )] ≤ PB (n, m, )

 m   e     1 n e e A w n . +  (1 − )n−e min 1, q − 1 w=1 w w e e=1 (23) Proof: Equation (23) is a simple consequence of Theorem (3). It is important to underline that as opposed to (14) and (9), which hold for specific parity-check ensembles, the upper union bound (23) holds for any parity-check ensemble whose expected WEF is known. Indeed, the upper bounds (14) and (9) may be derived from (23) using the expected WEF for the ensembles C(n, m, q) and S(n, m, q, p), respectively. As illustrated in the following example, the upper bound (23) may be used to predict the performance of q-ary LDPC codes over the q-EC, under ML decoding. Example 3: Consider Gallager L(n, J, K, q) regular LDPC ensemble, where J is the variable node degree and K is the check node degree. The WEF of a degree-K check node is given by3 a(x) =

The quantity Pr{∪x∈G {supp(x) ⊆ IE }| |IE | = e} in (21) may be now upper bounded as Pr{ ∪x∈G {supp(x) ⊆ IE }| |IE | = e} 

 Pr{supp(x) ⊆ IE | |IE | = e} (a) ≤ min 1, |Sx | 

x∈G  Pr{supp(x) ⊆ IE | |IE | = e} (b) ≤ min 1, q−1

x∈G    e  e Aw 1 (c) n . = min 1, (22) w w q−1 w=dmin

In the previous equation, (a) follows from the union bound and from the fact that for a linear block code over Fq a multiplicity of codewords share the same support so that, if x and x are any two such codewords, we have {supp(x) ⊆ IE } ∪ {supp(x ) ⊆ IE } = {supp(x) ⊆ IE } . Moreover, (b) follows from Proposition 2 and (c) from Pr{supp(x) ⊆ IE | |IE | = e} = Pr{supp(x ) ⊆ IE | |IE | = e} for any two codewords x and x with the same Hamming weight. The statement follows by incorporating the upper bound (22) into (21). Theorem 3 can be exploited to obtain an upper bound on the expected performance over the q-EC of a q-ary code drawn randomly from any parity-check ensemble, as follows. Theorem 4: Let G(n, m, q) be a genericparity-check enn semble over Fq . Moreover, let A(x) = 1 + i=1 Ai xi be the expected WEF of a code G drawn randomly from G(n, m, q). Then, the expected block error probability of G, under ML decoding and over a q-EC with erasure probability , may be

1 q−1 (1 + (q − 1)x)K + (1 − x)K . q q

Form this result, the average WEF of L(n, J, K, q) can be expressed as   J  n n    (a(x)) K , xi n i coeff n xi A(x) = 1 + (q − 1) i i (q − 1) i i=2   where by coeff P (x), xi we denote the coefficient of xi in the polynomial P (x). The upper bound (23) is provided in Figure 5 for various (120, J, K, 4) Gallager ensembles. The tightness of the upper bounds has been confirmed through an extensive Monte Carlo simulation campaign. For each ensemble, 104 codes have been randomly picked and their block error rates over the 4-ary erasure channel have been measured under ML decoding and averaged over the sample. The resulting averaged error rates are reported in Figure 5, and show to closely approach the corresponding union bounds (23). Figure 6 depicts the union bound (23) for L(80, 5, 10, q) regular LDPC ensembles over F2 , F4 , and F16 . On the same plot, the Singleton bound for n = 80 and m = 40 is depicted together with the upper bound (14) on the average block error probability for the C(80, 40, q) parity-check ensemble with q = 2, 4, and 16. Already with q = 16, the random coding bound performance in almost undistinguishable from the Singleton bound, down to the considered block error probability. The LDPC ensemble upper bounds (23) tightly follow the corresponding random coding bounds for q ≥ 4 3 The derivation of a(x) is provided in [8]. Alternatively, it can be derived by the WEF associated with a repetition code of length K over Fq , b(x) = 1 + (q − 1)xK , via the MacWilliams identity [40] as   1 1−x a(x) = (1 − (q − 1)x)K b . q 1 + (q − 1)x

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down to moderate block error probability. At lower error rates, the average LDPC ensemble performance deviates remarkably from the random coding bounds, showing how on average the LDPC ensembles have error floors. Here, the average performance is dominated by the low minimum distance of the worse ensemble members. For larger field orders, the floor appears at much lower block error probabilities. In the specific case, the expected error floor is lowered from block error probability 10−4 to 10−7 when moving from q = 4 to q = 16. Therefore, codes drawn from L(80, 5, 10, 16) can be used to attain a nearly-optimum performance on a 16-ary erasure channel down to low error probabilities and with a manageable decoding complexity thanks to efficient ML decoders [11], [12], [18], [41]. This consideration can be extended in general to the L(n, J, K, q) ensemble down to even smaller n and moderate q. Finally, in Figure 7, the bounds (19) and (23) are compared for the L(80, 5, 10, q) regular LDPC ensembles for q = 4 and q = 16. The bound (23) is appreciably tighter. V. C ONCLUSIONS In this paper tight upper bounds on the average block error probability for several code ensembles have been derived, over the q-EC. A comparison between an extension of the Berlekamp random coding bound and the Gallager random coding bound on the q-EC has been provided, demonstrating how the extended Berlekamp bound reveals to be tighter down to low error probabilities. Hence, it can be used to tightly estimate the performance achievable by the best (not necessarily linear) block code. Tight upper bounds on the average block error probability for ensembles characterized by sparse parity-check matrices have also been developed, showing how a nearly-ideal performance can be achieved by sparse-graph codes, down to low error probabilities, already for moderate field orders and short blocks (where usually binary codes show visible performance losses w.r.t. the Singleton bound).

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ACKNOWLEDGMENT We wish to thank the Associate Editor for his efficient handling of the review process, and the Anonymous Reviewers whose detailed comments helped to improve the readability of this paper. A PPENDIX A U NIFORM PARITY-C HECK E NSEMBLE OVER B LOCK E RASURE C HANNELS A memoryless L-BLEC is an erasure channel in which bits are grouped in blocks of L bits, and each block is erased

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independently of the other blocks with some probability . We refer to each block as a supersymbol [21]. Clearly, an L-BLEC can be seen as a q-EC with q = 2L , or as L/l fully-correlated [30] q-ECs with q = 2l (where l divides L). Next, we consider the problem of characterizing the expected performance under ML decoding of a linear block code G drawn from the uniform parity-check ensemble C(n, m, 2l ), for integer L/l, ns = n l/L, and ms = m l/L. Under the above assumptions, a Berlekamp upper bound on the average block error probability of the C(n, m, 2l ) ensemble is given by (S)

EC(n,m,2l ) [PB (G, )] < PB (ns , ms , ) ms   ns e 1  + l  (1 − )ns −e 2−L(ms −e) . 2 − 1 e=1 e

(24)

Figure 8 depicts the upper bound on the block error probability of parity-check ensembles over F2 , F4 , F16 , and F256 , for an 8-BLEC. The ensemble parameters are n = ns L/l and m = ms L/l with ns = 16, ms = 8, L = 8, and l = log2 q. As a result, all ensembles share the same coded block length in bits, nb = 128. Note that also in this setting the adoption of a higher finite field order allows to obtain a performance tightly approaching the Singleton bound. R EFERENCES [1] P. Elias, “Coding for two noisy channels,” in Proc. 1955 Inf. Theory: Third London Symp., pp. 61–74. [2] M. Luby, M. Mitzenmacher, A. Shokrollahi, D. A. Spielman, and V. Stemann, “Practical loss-resilient codes,” in Proc. 1997 ACM Symp. Theory Comput., pp. 150–159. [3] J. Byers, M. Luby, and M. Mitzenmacher, “A digital fountain approach to reliable distribution of bulk data,” IEEE J. Sel. Areas Commun., vol. 20, no. 8, pp. 1528–1540, Oct. 2002. [4] D. Lun, M. M´edard, R. Koetter, and M. Effros, “On coding for reliable communication over packet networks,” Physical Commun., vol. 1, no. 1, pp. 3–20, Mar. 2008. [5] M. Luby, M. Mitzenmacher, A. Shokrollahi, and D. A. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 585–598, Feb. 2001.

[6] M. Luby, “LT codes,” in Proc. 2002 IEEE Symp. Foundations Comput. Science, pp. 271–282. [7] M. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, no. 6, pp. 2551–2567, June 2006. [8] R. G. Gallager, Low-Density Parity-Check Codes. M.I.T. Press, 1963. [9] H. D. Pfister, I. Sason, and R. Urbanke, “Capacity-achieving ensembles for the binary erasure channel with bounded complexity,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2352–2379, July 2005. [10] M. Lentmaier, A. Sridharan, D. Costello, Jr., and K. Zigangirov, “Iterative decoding threshold analysis for LDPC convolutional codes,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 5274–5289, Oct. 2010. [11] H. Pishro-Nik and F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 50, no. 3, pp. 439–454, Mar. 2004. [12] D. Burshtein and G. Miller, “An efficient maximum likelihood decoding of LDPC codes over the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 50, no. 11, pp. 2837–2844, Nov. 2004. [13] B. Vellambi and F. Fekri, “Results on the improved decoding algorithm for low-density parity-check codes over the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 53, no. 4, pp. 1510–1520, Apr. 2007. [14] E. Paolini, G. Liva, B. Matuz, and M. Chiani, “Generalized IRA erasure correcting codes for hybrid iterative/maximum likelihood decoding,” IEEE Commun. Lett., vol. 12, no. 6, pp. 450–452, June 2008. [15] V. Roca, C. Neumann, and D. Furodet, “Low density parity check (LDPC) staircase and triangle forward error correction (FEC) schemes,” Request for comment 5170 (Standards Track/Proposed Standard), IETF RMT Working Group, June 2008. [16] M. Cunche, V. Savin, and V. Roca, “Analysis of quasi-cyclic LDPC codes under ML decoding over the erasure channel,” in Proc. 2010 Int. Symp. Inf. Theory Appl., pp. 861–866. [17] B. Schotsch, R. Lupoaie, and P. Vary, “The performance of low-density random linear fountain codes over higher order Galois fields under maximum likelihood decoding,” in Proc. 2011 Allerton Conf. Commun., Control, Comput., pp. 1004–1011. [18] E. Paolini, G. Liva, B. Matuz, and M. Chiani, “Maximum likelihood erasure decoding of LDPC codes: pivoting algorithms and code design,” IEEE Trans. Commun., vol. 60, no. 11, pp. 3209–3220, Nov. 2012. [19] E. Berlekamp, “The technology of error-correcting codes,” Proc. IEEE, vol. 68, no. 5, pp. 564–593, May 1980. [20] S. MacMullan and O.M.Collins, “A comparison of known codes, random codes, and the best codes,” IEEE Trans. Inf. Theory, vol. 44, no. 7, pp. 3009–3022, Nov. 1998. [21] A. Guillen i Fabregas, “Coding in the block-erasure channel,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 5116–5121, Nov. 2006. [22] C. Di, D. Proietti, T. Richardson, E. Telatar, and R. Urbanke, “Finite length analysis of low-density parity-check codes on the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1570–1579, June 2002. [23] B. Schotsch, H. Schepker, and P. Vary, “The performance of short random linear fountain codes under maximum likelihood decoding,” in Proc. 2011 IEEE Int. Conf. Commun., pp. 1–5. [24] A. Bennatan and D. Burshtein, “On the application of LDPC codes to arbitrary discrete-memoryless channels,” IEEE Trans. Inf. Theory, vol. 50, no. 3, pp. 417–438, Mar. 2004. [25] E. Hof and I. Sason and S. Shamai, “Performance bounds for nonbinary linear block codes over memoryless symmetric channels,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 977–996, Mar. 2009. [26] A. Bennatan and D. Burshtein, “Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 549–583, Feb. 2006. [27] D. Lucani, M. M´edard, and M. Stojanovic, “Random linear network coding for time-division duplexing: field size considerations,” in Proc. 2009 IEEE Global Telecommun. Conf., pp. 1–6. [28] G. Liva, E. Paolini, and M. Chiani, “Performance versus overhead for fountain codes over Fq ,” IEEE Commun. Lett., vol. 14, no. 2, pp. 178– 180, Feb. 2010. [29] C. Koller, M. Haenggi, J. Kliewer, and D. Costello, Jr., “On the optimal block length for joint channel and network coding,” in Proc. 2011 IEEE Inf. Theory Workshop, pp. 528–532. [30] J. Massey, “Capacity, cutoff rate, and coding for a direct-detection optical channel,” IEEE Trans. Commun., vol. 29, no. 11, pp. 1615–1621, Nov. 1981. [31] R. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory, vol. 27, no. 4, pp. 393–398, July 1981. [32] S. Wilson, Digital Modulation and Coding. Prentice Hall, 1995. [33] S. Fashandi, S. O. Gharan, and A. K. Khandani, “Coding over an erasure channel with a large alphabet size,” in Proc. 2008 IEEE Int. Symp. Inf. Theory, pp. 1053–1057.

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[34] R. C. Singleton, “Maximum distance q-nary codes,” IEEE Trans. Inf. Theory, vol. 10, no. 2, pp. 116–118, Apr. 1964. [35] R. G. Gallager, Information Theory and Reliable Communication. Wiley, 1968. [36] D. Divsalar, S. Dolinar, and C. Jones, “Protograph LDPC codes over burst erasure channels,” in Proc. 2006 IEEE Military Commun. Conf., pp. 1–7. [37] R. Lidl and H. Niederreiter, Finite Fields. Cambridge University Press, 1997. ¨ [38] G. Lansberg, “Uber Eine Anzahlbestimmung und eine damit zusammenh¨angende Reihe,” J. f¨ur die Reine Angewandte Mathematik, vol. 3, pp. 87–88, 1893. [39] A. Terras, Fourier Analysis on Finite Groups and Applications. Cambridge University Press, 1999. [40] F. Mac Williams and N. Sloane, The Theory of Error-Correcting Codes. North Holland Mathematical Libray, 1977, vol. 16. [41] G. Garrammone and B. Matuz, “Short erasure correcting LDPC IRA codes over GF(q),” in Proc. 2010 IEEE Global Telecommun. Conf., pp. 1–5. Gianluigi Liva (M’08) was born in Spilimbergo, Italy, on July 23rd, 1977. He received the M.S. and the Ph.D. degrees in electrical engineering from the University of Bologna (Italy) in 2002 and 2006, respectively. His main research interests include satellite communication systems, random access techniques and error control coding. Since 2003 he has been involved in the research of channel codes for high data rate CCSDS (Consultative Committee for Space Data Systems) missions, in collaboration with the European Space Operations Centre of the European Space Agency (ESAESOC). From October 2004 to April 2005 he was researching at the University of Arizona as visiting student, where he was involved in the design of low-complexity coding systems for space communication systems. He is currently with the Institute of Communications and Navigation at the German Aerospace Center (DLR). He is / he has been active in the DVB-SH, DVB-S2 and in the DVB-RCS standardization groups. In 2010 he has been appointed guest lecturer for channel coding at the Institute for Communications Engineering (LNT) of the Technische Universit¨at M¨unchen (TUM). In 2012 and 2013 he has been lecturing for channel coding at the Nanjing University of Science and Technology in Changshu (China). Dr. Liva is IEEE member and he serves IEEE as reviewer for Transactions, Journals and Conferences. Since 2013 he is Editor for IEEE C OMMUNICA TIONS L ETTERS .

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Enrico Paolini (M’08) was born in Fano, Italy, in 1977. He received the Dr. Ing. degree (with honors) in Telecommunications Engineering in 2003 and the Ph.D. degree in Electrical Engineering in 2007, both from the University of Bologna, Italy. While working toward the Ph.D. degree, he was Visiting Research Scholar with the University of Hawai’i at Manoa. From 2007 to 2010, he held a postdoctoral position with the Department of Electronics, Computer Science and Systems of the University of Bologna. Currently, he is an Assistant Professor at the Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi”, University of Bologna. His research interests include error-control coding (with emphasis on LDPC codes and their generalizations, iterative decoding algorithms, reduced-complexity maximum likelihood decoding for packet erasure channels), random access techniques, and radar sensor networks based on UWB. In the field of error correcting codes, has been involved since 2004 in several activities with the European Space Agency (ESA). In the summer 2012 he was Visiting Scientist at the Institute of Communications and Navigation of the German Aerospace Center. Dr. Paolini is Editor for the IEEE C OMMUNICATIONS L ETTERS . He served on the Technical Program Committee at several IEEE International Conferences, and on the Organizing Committee (as treasurer) of the 2011 IEEE International Conference on Ultra-Wideband. He is member of the IEEE Communications Society and of the IEEE Information Theory Society. Marco Chiani (M’94-SM’02-F’11) received the Dr. Ing. degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic and computer engineering from the University of Bologna, Italy, in 1989 and 1993, respectively. He is a Full Professor in Telecommunications and the current Director of the Industrial Research Center on ICT at the University of Bologna. During summer 2001, he was a Visiting Scientist at AT&T Research Laboratories, Middletown, NJ. Since 2003 he has been a frequent visitor at the Massachusetts Institute of Technology (MIT), Cambridge, where he presently holds a Research Affiliate appointment. He is leading the research unit of the University of Bologna on cognitive radio and UWB (European project EUWB), on Joint Source and Channel Coding for wireless video (European projects Phoenix-FP6, Optimix-FP7, Concerto-FP7), and is a consultant to the European Space Agency (ESA-ESOC) for the design and evaluation of error correcting codes based on LDPCC for space CCSDS applications. His research interests include wireless communication systems, MIMO systems, wireless multimedia, error correcting codes, cognitive radio and UWB. He recently received the 2011 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems, the 2012 IEEE Communications Society Fred W. Ellersick Prize, and the 2012 IEEE Communications Society Stephen O. Rice Prize in the Field of Communications Theory. He is the past chair (2002-2004) of the Radio Communications Committee of the IEEE Communication Society and past Editor of Wireless Communication (2000-2007) for the journal IEEE T RANSACTIONS ON C OMMUNICATIONS. Since 2011 he is a Fellow of the IEEE, named for “Contributions to wireless communication systems”. He is a Distinguished Lecturer for the IEEE ComSoc (2011/2012). In 2012 he has been appointed Distinguished Visiting Fellow of the Royal Academy of Engineering, UK.