Erasure correction performance of linear block codes

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V olume 781, 1994, DOI: 10.1007/3-540-57843-9

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First French-Israeli Workshop Paris, France, July 19–21, 1993 Proceedings

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G. Cohen, S. Litsyn, A. Lobstein and G. Zémor Fr o nt m atter A necessar y and sufficient

1-10

co nditio n fo r tim e-v ar iant co nv o lutio nal enco der s to be no ncatastr o phic On the desig n and selectio n

11-21

o f co nv o lutio nal co des fo r a bur sty

Contents V iew ing all 32 c hapters

Rician channel M o dulo -2 separ able linear

22-27

co des Estim atio n o f the size o f the

28-33

list when deco ding o v er an ar bitr ar ily v ar ying channel

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A necessary and sufficient condition for time-variant convolutional encoders to be noncatastrophic

1-10

V. B. Balak ir sk y

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On the design and selection of convolutional codes for a bursty Rician channel

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G ideo n Kaplan, Shlo m o Sham ai and Yo sef Ko fm an


Modulo-2 separable linear codes

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G r eg o r y Po ltyr e v and Jak o v Snyder s


Estimation of the size of the list when decoding over an arbitrarily varying channel

Show Summary 28-33

V. Blino v sk y and M . Pinsk er


A lower bound on binary codes with covering radius one

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Iir o Ho nk ala


On some mixed covering codes of small length

Show Summary 38-50

E. Ko lev and I. Landg ev


The length function: A revised table

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A nto ine Lo bstein and Ver a Pless


On the covering radius of convolutional codes

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Ir ina. E. Bo char o v a and Bo r is. D. Kudr yasho v


Efficient multi-signature schemes for cooperating entities

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Oliv ier Delo s and Jean-Jacques Quisquater


Montgomery-suitable cryptosystems

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SpringerLink - Contents Dav id Naccache and Dav id M 'Raïhi


Secret sharing schemes with veto capabilities

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C. Blundo , A . De Santis, L. G ar g ano and U. Vaccar o


Group-theoretic hash functions

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Jean-Pier r e Tillich and G illes Zém o r


On constructions for optimal optical orthogonal codes

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Sar a Bitan and Tuv i Etzio n


On complementary sequences


A m no n G av ish and A br aham Lem pel


Spectral-null codes and null spaces of Hadamard submatrices


Ro n M . Ro th


On small families of sequences with low periodic correlation


Sascha Bar g


Disjoint systems (Extended abstract)


No g a A lo n and Benny Sudak o v


Some sufficient conditions for 4-regular graphs to have 3-regular subgraphs


Oscar M o r eno and Victo r A . Zino v iev


Detection and location of given sets of errors by nonbinary linear codes


M ar k G . Kar po v sk y, Saeed M . Chaudhr y, Lev B. Lev itin and Claudio M o r ag a


Quaternary constructions of formally self-dual binary codes and unimodular lattices


A lexis Bo nnecaze and Patr ick So lé


New lower bounds for some spherical designs


Peter Bo yv alenk o v and Sv etla Nik o v a


Lattices based on linear codes


G r eg o r y Po ltyr e v


Quantizing and decoding for usual lattices in the Lp-metric


P. Lo yer and P. So lé


Bounded-distance decoding of the Leech lattice and the Golay code


Ofer A m r ani, Yair Be'er y and A lexander Var dy


Some restrictions on distance distribution of optimal binary codes


Ser g ei I. Ko v alo v


Two new upper bounds for codes of distance 3



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SpringerLink - Contents Sim o n Litsyn and A lexander Var dy



On Plotkin-Elias type bounds for binary arithmetic codes G r eg o r y Kabatiansk i and A nto ine Lo bstein



Bounds on generalized weights G ér ar d Co hen, Llo r enç Hug uet and G illes Zém o r



Threshold effects in codes G illes Zém o r



Decoding a bit more than the BCH bound Jo sep Rifà Co m a



Product codes and the singleton bound Nico las Sendr ier



Erasure correction performance of linear block codes Ilya I. Dum er and Patr ick G . Far r ell


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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, A g E * 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. . . . . , a0 = uG. Let J = {.Jl. . . . . j,} denote an ordered set of s unerased positions in the received vector z = (zl, ..., z,) E Sn, 1 (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 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

o f 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 ctt 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. proposition 1. QED.

Now the proof follows from

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,--~ Wi(;)/(;)

(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; when n ~ 00.

o(1)--,0,

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 - l o g ( 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+%)

for all d* ~ i < n-d*, where en

=

(9)

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 ~ m s - o(m) where o(m) is any positive function increasing more slowly than m.

(10)

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 KarlitsUchiyama 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{ ( ; ) / ( ; ) } , w h i c h

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

Erasure correction performance of linear block codes

Erasure correction performance of linear block codes

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