Erasure Correction Performance of Linear Block Codes.

Abstract

We estimate the probability of incorrect decoding of a linear block code, used over an erasure channel, via its weight spectrum, and define the weight spectra that allow us to achieve the capacity of the channel and the random coding exponent. We derive the erasure correcting capacity of long binary BCH codes with slowly growing distance and their duals. Concatenated codes of growing length n and polynomial decoding complexity O(n2), achieving the capacity of the erasure channel (or any other discrete memoryless channel), are considered.

Full text

COMPUTER SCIENCE
LECTURE NOTES IN COMPUTER SCIENCE
Volume 781, 1994, DOI: 10.1007/3-540-57843-9
Algebraic Coding
First French-Israeli Workshop Paris, France, July 19–21, 1993 Proceedings
G. Cohen, S. Litsyn, A. Lobstein and G. Zémor
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A necessary and sufficient condition for time-variant convol utional encoder s to be
noncatastr ophic
V. B. Balakirsky
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11-21
On the design and selection of convolutional codes for a bursty Rician channel
Gideon Kaplan, Shlomo Shamai and Yosef Kofman
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22-2
7
Modul o-2 separ able linear codes
Greg ory Poltyrev and Jakov Snyder s
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28-33
Estimation of the size of the list when decoding over an arbitrarily varying channel
V. Blinovsky and M. Pinsker
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34-3
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A l ower bound on binary codes with cover ing radius one
Iiro Honkala
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38-5
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On some mixed covering codes of small l ength
E. Kole v and I. Landg ev
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51-55
The length function: A revised table
Antoine Lobstein and Vera Pless
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56-62
On the covering radius of convolutional codes
Irina. E. Bocharova and Bor is. D. Kudryashov
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63-7
4
Efficient mul ti-signature schemes for cooperating entities
Olivier Delos and Jean-Jacques Quisquater
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75-81
Montgomer y-suitable cryptosystems
Front m atter
1-10
A necessary and sufficient
condition for time-variant
convolutional encoders to be
noncatastrophic
11-21
On the design and selection
of convolutional codes for a bursty
Rician channel
22-27
Modulo-2 separable linear
codes
28-33
Estimation of the size of the
list when decoding over an
arbitrarily varying channel
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David Naccache and David M'Raïhi
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Secret sharing schemes with veto capabilities
C. Blundo, A. De Santis, L. Gargano and U. Vaccaro
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Group-theoretic hash functions
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On constructions for optimal optical orthogonal codes
Sara Bitan and Tuvi Etzion
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On complementary sequences
Amnon Gavish and Abraham Lempel
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141-153
Spectral -null codes and null spaces of Hadamard submatrices
Ron M. Roth
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154-158
On small families of sequences with low periodic correlation
Sascha Barg
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Disj oint systems (Extended abstr act)
Noga Alon and Benny Sudakov
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Some sufficient conditions for 4-regular gr aphs to have 3-regular subgr aphs
Oscar Moreno and Victor A. Zinoviev
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Detection and location of given sets of er rors by nonbinary linear codes
Mark G. Karpovsky, Saeed M. Chaudhry, Lev B. Levitin and Claudio M oraga
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194-205
Quaternar y constructions of formall y self-dual binar y codes and unimodular l attices
Alexis Bonnecaze and Patrick Solé
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207-216
New l ower bounds for some spherical designs
Peter Boyvalenkov and Svetla Nikova
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Lattices based on linear codes
Gregory Poltyr ev
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Quantizing and decoding for usual l attices in the
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P. Loye r and P. Solé
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Bounded-distance decoding of the Leech lattice and the Golay code
Ofer Amrani, Yair Be'ery and Alexander Vardy
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Some restrictions on distance distr ibution of optimal binary codes
Sergei I. Kovalov
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Two new upper bounds for codes of distance 3
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Threshold effects in codes
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Product codes and the singl eton bound
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Erasure correction performance of linear block codes
Ilya I. Dumer and Patrick G. Farrell
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Erasure Correction Performance
of Linear Block Codes
Ilya I. Dumer* and Patrick G. Farrell**
* Institute for Problems of Information Transmission
Moscow, Russia
** University of Manchester
United Kingdom
Abstract
We estimate the probability of incorrect decoding of a linear block code, used over an erasure
channel, via its weight spectrum, and define the weight spectra that allow us to achieve the
capacity of the channel and the random coding exponent. We derive the erasure correcting
capacity of long binary BCH codes with slowly growing distance and their duals.
Concatenated codes of growing length n --* 0o and polynomial decoding complexity O(n2),
achieving the capacity of the erasure channel (or any other discrete memoryless channel), are
considered.
1. Introduction
In this paper we consider the performance of binary linear block codes used over an erasure
channel. The channel is defined by the input set E = {0,1}, the output set S = {0,1,*},
including the erasure symbol *, and the set of transition probabilities
P(*I 0) -- P(*I 1)
= p,
and p(0/0) = p(1 [ 1) = 1-p. The capacity of the erasure channel, C = l-p, and the random
coding exponent, E(R,p) for codes of rate R are well known (see [1]).
Consider a binary n-dimensional Hamming space E ~ = {0,1} n and let A, AgE * be a linear
(n,k)-code of length n and code rate R = k/n, defmed by a (kxn) generator matrix G = (&j),
i = 1 ..... k; j = 1 ..... n. The input information sequence u= (ul, u: ..... u~) is encoded into
the codeword a = (al, a2 ..... , a 0 = uG. Let J = {.Jl ..... j,} denote an ordered set of s
unerased positions in the received vector z = (zl, ..., z,) E S n, 1 <-Jl -<... J, -< n. Hereafter
zj = (zjl j E J) E E' denotes the unerased s-subvector of vector z and Gj = (glj), i = 1 .....
k; j EJ , denotes a (kxs)-sub matrix of G, defined on the set J. Obviously z~ = at.
The maximum likelihood (ML) decoder derives the information sequence u E E k by solving
the system.
uGj = zj (1)
of s linear equations in k binary variables u 1 ..... u k. Note that system (1) has a unique
solution, and decoding is unambiguous, if and only if the rank r(J) = Rk(Gj) of matrix Gj
is equal to k. Otherwise any solution among the set Of L = 2 k<J) solutions is possible and the
probability of correct decoding is equal to 1/L. ML-decoder (1) is known to provide the
minimal probability of incorrect decoding (per block) of any linear code A with polynomial
complexity O(n3), n --,
oo.
Let ett denote the fraction of erasure patterns of weight t resulting in ambiguous decoding, for

317
a given code A with code distance d. The probability P of ambiguous decoding is defined by
the set {c~:, t=d, ...,n-k} sincec~t = 0 for t < dandc~ = 1 for t > n-k. Below we
consider so called (n,R)-families of (ni,lq). -codes, with rates ~ = ki./nl, converging to R,
0 < R < 1, when i --} oo. We prove further, that the set of inequalities off < oe 2 ~ .... < or, holds
for any linear code and consider (n,R) - families with the following threshold property.
Definition 1 The infinite (n,R) - family is called an (n,R,O)-family, 0 < 0
~1, if for any E>0 and any ti_<(1-~)0(n:lq), the fraction ~tt--'0, when i--}oo, while for any
ti>(1 +E)0(n:lq). the fraction 0ctt > 3' for some 3'>0, when i --* oo.
According to proposition 1 below, the parameter 0 defines the relative erasure
capacity
of the
(n,R)-family. Namely, an (n,R,0)-family gives a vanishing probability of incorrect decoding,
if used over an erasure channel with transition probability p<0(1-R). We are mainly
interested in constructing (n,R,1)-families of codes, since only these families achieve the
capacity 1-p of the erasure channel for any R< l-p, and all 0<p<l.
In section 2 we estimate the numbers c~ t and probability P of ambiguous decoding via the
weight spectrum W = (1,W d ..... W,) of code A, where Wl,i=d ..... n, is the number of
codewords of Hamming weight i. We prove that an (n,R,1)-family is obtained, if the
restrictions
hold. Moreover, the probability P decreases exponentially with distance d for any p < 1-R.
Restrictions (2) allow any growth of distance d, no matter how slow it is.
On the other hand, much stronger restrictions Wl < Li =+~),
where i = 1 ..... n(1-R), o(1)--,0, n --} oo and
{ (7) / 2"O-R), l < i<n (1-R)/2
LI = (;)/(n0;R)),
/I(1-R)
/ 2 ~; I ~ n(1-R)
(3)
allow us to achieve the random coding exponent E(R,p), 0 < R < 1-p. Moreover, the exponent
E(R,p) can be achieved, for any p < I-R, only if the distance d satisfies the asymptotical

318
inequality d/n ~ 8, n --* oo, ~5 being the relative Varshamov-Gilbert distance of the code;
= H-I(1.R), 8 < t~ and H(x) = - x log2x - (l-x) log 2 (l-x) is the binary entropy function,
0~x-:: 1
In section 3 we consider the erasure capacity of primitive BCH-codes of growing length
n=2m-l~oo and fixed or slowly growing distance. We prove that these codes correct virtually
all erasure patterns of weight t ~ em = n-k, where e = L(d-1)/2.]. Therefore these codes
make up an (n, 1, l)-family. We also prove that their dual codes correct virtually all erasure
patterns of weight t - n-era and therefore form an (n,0,1)-family.
Following Forney [2], we consider in section 4 the (n,R)-families of concatenated codes and
define the (n,R, 1)-families with maximal possible erasure capacity and complexity of order
O(n 2) for any 0<R< 1 and n-~. These families provide a positive answer to an interesting
question brought up by Zemor and Cohen [3]. Namely, codes satisfying the weight
restrictions in inequality(3), are proved in [3] to achieve the maximal capacity 0= 1, and the
question raised there is whether any constructive families achieve the same capacity.
In section 5 we state some open problems resulting from our investigation.
2. Erasure Performance of Linear Block Codes
Consider a binary linear (n,k)--code A, used over a binary erasure channel with erasure
probability p. Let the codeword positions be N={I,2 ..... n}. I = {il, i2 ..... it} is a set of t
erased positions, and J = N\I is the complementary set of n - t unerased positions. For any
codeword b = (b 1 ... b~) define the set Supp (b) = {j[bj = 1}. The set I is defined as a
covering set, iff (if and only if) Supp (b) c I for at least one non-zero codeword a E A. An
ML-decoder tries to restore the transmitted codeword a from its unerased subvector aj = zj
by solving the system (1).
/.,emma 1. The following conditions (i), (ii), (iii) are equivalent:
(i)
(2)
(iii)
The decoding of erasure pattern I is ambiguous
I is a covering set
The rank r(J) of the matrix Gj is less than k
The proof is obvious.
Let M, = {I:}I1 =t, r(J)<k) denote the complete set of covering t-subsets. Let o~ =
[Mill(: ) denote the fraction of covering t-subsets among aU (7)subsets ofcardinality t.
Lemma 2. The numbers c~ t form a nondecreasing l~anction of the weight t.
Proof: any coveting t-subset can be obtained from t+ 1 covering subsets of weight t at most.
Therefore
IM,+,I > IM, I.
(n-t)/(t+l)and c~,+~ > at. QED

319
According to Lemma 1, the probability P of ambiguous decoding is
n n
e -- M, p'
(X-p)"-' --
t=d t=d
(4)
where/3(0 = (7)pt (1.p),~ is the binomial distribution of the probability oft erasures on n
positions. The probability P, of incorrect decoding can be estimated as: P/2 g Pe < P, since
the transmitted eodeword is selected from among 2 k'roJ > 2 possible codewords whenever the
decoding is ambiguous. Thus the problem of estimating the probabilities P and Pe leads to the
problem of estimating the numbers c~.
Proposition I. Any (n,R,0)-family with 0 < R<I has vanishing probability P -* 0 of
ambiguous decoding for any p < 0(l-R) and non-vanishing probability P > 1' for any p >
0(l-R), and some 3'>0,when n ~ oo.
Proof: according to definition 1, at ~ 0 for any ~ > 0 and t<r = L(1-~)0(1-R)nJ, when
n ~ oo. From I..emma 2 the inequality follows:
'~ /I II
t=d
r247 t=T§
For p < 0(l-R) and E small enough, the inequality T/n>p holds and the probability
n
of r 1 or more erasures tends to zero, when n -* o0. Since a, ~ 0, the first part of the
proposition follows. On the contrary, for any ~ > 0 and t ~ T = L(I+~)0(1-R)nJ the
inequalities at > 3' and
1 (0
t=T
hold.

320
/I
For p > 0(l-R) and ~ small enough E
T
follows. QED
~(t) --, 1, and the second part of the proposition
According to Proposition 1 (n,R,0)-families achieve the capacity 1-p of the erasure channel
iff 0= 1. Hereafter we consider (n,R, 1)-families and estimate their weight spectra.
Proposition 2. If the weight spectra of the (n,R)-family satisfy restrictions (2), then the family
achieves the maximal erasure capacity 0= 1 and has probability P of ambiguous decoding,
decreasing exponentially with distance d --, oo for any p < 1-R.
Proof : following [3], we estimate the numbers ct t via the weight spectrum Wi, i=d ..... n-k.
Let ~t (i) be the fraction of t-subsets covering codewords of weight i, t > i. Obviously,
t
.., = .., (l)/(;)
i=d
since each codeword of weight i is covered by
(a-i)t_i
subsets of weight t.
If restrictions (2) are satisfied, then
9 (;) (7)
Therefore ,~t(i) < t ~ and c~, < f~/(1-f), where f = t/(n-k).
For any e > 0 and t < (1-~)(1-R)n the estimate
=, < (1-~)d/~
holds and therefore ~xt decreases exponentially with d. Now the proof follows from
proposition 1. QED.
More explicit estimates of the numbers ,, (see (5)) can be obtained in the following way. Let
c~,j denote the fraction of t-subsets covering ~ codewords, j=l ..... t. Obviously
t ! t
j=l 1=1 i=d

321
since any t-subset, covering 2J-1 non-zero codewords, is counted 2J-1 times in the right hand
side of the last equality. Let D = {ds=d, d 2 ..... dk=n } denote the set of generalised
minimum distances [10] of a linear (n,k)-code. Any t-subset, d I < t < d 2, covers at most
one non-zero codeword and the equality
0c,--~ W i(;)/(;) (6)
i--d
holds for these t. For larger values of t lower estimates of ~ can be obtained, if the sets D
and W are known.
Hereafter all unspecified logarithms and exponents are defined over binary base;
o(1)--,0,
when n ~ 00.
Consider now the weight restrictions of (n,R)-families that give the random coding exponent
E(R,p) under ML decoding for all 0 < p < I-R. Let T(3') = --flogp-(l -'y)log(l-p)-H('y)
denote the limiting exponent of the probability//(.rn) of erasure patterns of weight
vn, n
-,
00.
According to [1], the best families of codes satisfy the inequality (-log P)/n > E (R,p) + o(1)
for the probability P of ML-decoding, where n --, 0o and
IT(l-R), /f (l-R) / (1 +R)< p ~ (l-R)
E(R,p)
= t-log (l+p)
+ l-R, if p < (1-R)/(I+R)
(7)
Proposition 3
1. The random coding exponent E(R,p) is achieved for all p, 0 < p < l-R, by an
(n,R)-family, if the weight spectra satisfy the restrictions (3).
2. The random coding exponent E(R,p) is achieved for all p, 0 < p < l-R, by an
(n,R)-family, only if the code distances d(n) in the (n,R)-family satisfy the restrictions
d(n)/n > H'I(1-R) + o(1), n ~ 00.
Proof: similarly to proposition 2.
Consider now the set of codes, generated by (kxn)-matrices G = (&j), i = 1 ..... k, j = 1,
.... n, with rate R = k/n, 0 < R < 1. It is well known that virtually all matrices G have
rank k, when n --, 00, and generate (n,R)--codes, satisfying inequalities (3) for all i=l ..... n
(see[4]). Therefore we have:

322
Corollary 1.. Virtually all linear (n,R)-codes achieve the random coding exponent E(R,p) of
the erasure channel for any p < 1-R.
3. Performance of BCH-eodes over an erasure channel
Below we estimate the performance of long BCH-codes with slowly growing or fixed distance
used over an erasure channel. Similarly to the estimates of section 2, the performance can
be estimated via their weight spectra (whereas the performance in'a binary symmetric channel
is defined by the weight distribution of the coset leaders). Still not much is known about the
explicit weight spectra of algebraic constructions. The known results include [4] the weight
spectra of primitive BCH-codes correcting up to 3 errors, Reed-Muller codes of the second
order and some of their subcodes, and the weight spectra of the dual codes. Therefore the
performance of all these codes can be estimated from (5).
Let B(n,s) denote the primitive binary BCH-code of length n = 2m-1 and designed distance
d* = 2s + 1 witla k > n-ms information symbols. The asymptotical performance of B(n,s)-
codes with n ~ oo and slowly growing (or fixed) distance d* can be estimated by the
following result.
Lemma 4.[~1. The number W i of codewords of weight i in BCH-code B(n,s) with
m -* o% n = 2 = - 1, s < 0.2 {Ln(n)/ln(ln(n))}
(8)
can be estimated as
Wi=
((~)/2"-k)(1+%) (9)
for all d* ~ i < n-d*, where en
=
O(n'~
According to [4, section 9.3] the equality k=n-ms holds for long BCH-codes with parameters
(8). Therefore, unambiguous ML-decxxting of these codes in an erasure channel can be done
only if the number t of erased symbols satisfies the inequality t ~ ms - slog n, n --- oo.
Note also that the actual distance d of long BCH-eodes with parameters (8) coincides with the
designed distance d*. The following proposition gives estimates of numbers t~ t in the
asymptotical interval d* -: t ~ (n-k) (Do(l)), when n -* oo.
Proposition 4. BCH-codes with parameters (8) correct virtually all erasure patterns of weight
t ~ ms- o(m) (10)
where o(m) is any positive function increasing more slowly than m.

323
Corollary 2. BCH-codes with parameters (8) form an (n,l,1)-family.
Proof: by substituting the weight spectra coefficients (9) into (5).
Consider now the asymptoticai performance of the codes B~(n,s), dual to the codes B(n,s).
We estimate their performance under the following restrictions, with parameter c, O< c< 1:
m-,oo,n= 2 =-l,s < c2 L=teJ4 (11)
These restrictions are weaker than restrictions (8). Similarly to B(n,s)-codes, the relation k
= ms < ~(n) log n holds for long B ~ (n,s) codes. Moreover, according to the Karlits-
Uchiyama bound [4, section 9.9], the inequality:
d > 2 "~'- (s-l)2 ~ > 2~t(1-c) (12)
holds.
Proposition 5. B l (n,s)-codes with parameters (11) correct virtually all erasure patterns of
weight
t < n-k/(1-1og(l +c)) - o(m) (13)
where o(m) is any positive function, increasing more slowly than m.
Proof: according to inequality (5),
n n
,,,
(:), (;)- (;), (:)-- (:) / (:)
Consider the function f(t)= log{ (;)/(;)},which grows with t. Direct calculations
show that the asymptotic equality f(t) - (n - t) log (1 - d/n) holds when n ~ 00 and t - n.
According to (12), d/n > (1-c)/2. Therefore, the proposition holds since o~ t * 0 for n ~ 00,
and any t satisfying (13). QED
Corollary 3. The family of B l (n,s)-codes with parameters (I1) form an (n,O,l)-family.
Propositions 4 and 5 show that long BCH-codes B(n,s) with restrictions (8) and dual codes
B'(n,s) with restrictions (II) achieve the capacity 0=1, correcting Virtually all erasure
patterns of weight t - n-k, when n ~ 00. The problem of estimating the erasure correcting

324
capacity of BCH-codes of arbitrary rate R, 0 < R < 1, is still open. In the following section
we describe concatenated constructions achieving maximal possible capacity 0= 1 for any R,
0<R<I.
4. Concatenated Codes
We consider below the classical Forney construction of concatenated codes [2]. Let A(q,n,m)
denote a q-ary code of length n with m codewords A(i), i = 1,... ,m, and rate R A --- (logq m)/n.
Let B(m, e,d,M) denote an m-ary code of length e, Hamming distance d with M codeword and
rate R~ = (logmM)/e. The concatenated code C(q,N,M) of length N=ne and rate R=RAR B
is defined by replacing symbol ~, j=l ..... e in any codeword 01 ..... ie)EB by the
corresponding q-ary n-vector A(ij): (il,...,it) ..... (A(i 0 ..... A(i,)).
Consider the asymptotical performance of q-ary concatenated codes with fixed rate R, 0 <
R < 1, when n --, 0% e --, 00, in an arbitrary memoryless channel. We choose inner codes
A to be of very short length n=o(log N), which achieve the random coding exponent of the
channel and then we use ML-decoding matched to this channel. Families of B(m,e,d,M)-
codes with linearly growing distance for any rate R B < 1 and e ---, 0% are chosen as outer
codes regardless of the channel. Only bounded distance is required for the outer channel,
outer decoding providing correction of up to (d-l)/2 errors.
Such families of B(m,e,d,M)-codes themselves can be constructed as concatenated codes,
according to [6]. A complexity of construction, including encoding and decoding with
correction of up to (d-l)/2 errors, of O(I e) can be achieved [6].
Proposition 6. For any discrete memoryless channel with capacity C and any rate R, R <
C, there exist infinite families of concatenated (N,R)-eodes, N --, oo, with complexity O(FF)
of construction, encoding and decoding, that provide exponentially decreasing probability of
incorrect decoding with length N.
Proof." Similar to Forney's proof [2] for the binary symmetric channel.
Corollary 4. For any rate R, 0 < R < 1, there exist (N,R, 1)-families of concatenated codes
with complexity O(N2).
Notes.
1. An algebraic construction of concatenated codes, based on Justesen concatenated
codes, is considered in [7]. This construction provides the conditions of proposition
6 for so called regular channels, forming a subclass of discrete memoryless
symmetric channels. Searching algorithms are only used in [7] for constructing large
Galois fields, rather than for constructing good short codes, matched to the channel,
as above.
2. The exponent E(R) of an erasure channel can be achieved via a class of concatenated
codes in the following (non-constructive) way. Let A(2,n,m) be a binary code of
growing length n ~ oo and rate R^, satisfying the weight restrictions (3). Let B =
{B(m,e,d,M)} denote the ensemble of generalised Reed-Solomon codes of length

325
e=m-1 and rate R s. Consider the corresponding ensemble {C} of concatenated
codes of rate R=R^Rs. According to [8], virtually all codes from ensemble {C}
satisfy restrictions (3), if
R^ > 1 + log2 (1-H4(1-R)).
04)
Therefore according to proposition 2, we have:
Corollary 5. Virtually all linear concatenated codes C E {C} of length N --* 00 achieve random
coding exponent E(R,p) of the erasure channel for any 0 < p < l-R, under restriction (14).
3. Cascaded ML-decoding for the erasure channd is considered in [9] for concatenated
codes, where it is carried out by constructing lists of inner codewords for every
column. Though the maximal possible inner decoding list is estimated in [9] as the
complete list of M = 2 rma eodewords, the decoding complexity is claimed to be
O(N "2) for N --* 0o. We should note that the overall number of trials grows for this
algorithm as 2 NR, whenever complete lists of M codewords are considered.
Therefore the maximal complexity of the algorithm in [9] is actually determined by
an exhanstive search over all codewords. Therefore the problem of constructing ML-
decoding schemes for erasures of complexity O(N 2) is still open.
5. Concluding Remarks
The ML-decoding performance of linear block codes in an erasure channel is an important
problem from theoretical and practical points of view, since the decoding complexity is upper
bounded by polynomial order O(n 3) with length n, and because ereasure decoding forms the
basis of information set and covering algorithms for error corection. In this paper the erasure
decoding error probability is estimated via the weight spectrum. The estimates are
exponentiallyfight for virtually all codes and meet the random coding exponent of the erasure
channel. It is shown that long BCH codes with fixed or slowly growing distance and rate R
"* 1, and their duals with R --, 0, achieve asymptotically the maximal possible capacity of
erasure correction. The same holds for properly constructed concatenated codes of arbitrary
rate R, with polynomial complexity.
Finally, we state two open problems, that stem from our considerations:
1. What are the erasure capacities of important algebraic (n,R)-codes of rate 0<R< 1,
such as BCH codes and Reed-Muller codes?
2. Does the (n,R)-family 0<R<I, achieve the maximal possible capacity 1-R of
erasure correction, if the dual family achieves the capacity R?
Acknowledgement: The authors are grateful to the Royal Society, UK, for the financial
support that made this research possible.

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