Generalized weights and bounds for error probability over erasure channels

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1 Generalized weights and bounds for error probability over erasure channels Leandro Cruvinel Lemes Department of Electrical Engineering Federal University of Triangulo Mineiro Uberaba - MG, Brazil leandro@icteuftmedubr Marcelo Firer Department of Mathematics State University of Campinas Campinas - SP, Brazil mfirer@imeunicampbr Abstract New upper and lower bounds for the error probability over an erasure channel are provided, making use of Wei s generalized weights, hierarchy and spectra In many situations the upper and lower bounds coincide and this allows us to improve the existing bounds Results concerning MDS and AMDS codes are deduced from those bounds I INTRODUCTION Generalized weights of a linear code were introduced by Victor Wei in [1] and became relevant invariants in Coding Theory, being determined for particular classes of codes [2], [3], [4], [5], [6], [7], [8], [9] and bounded when explicit formulas are not available [1], [11] However, the importance of those invariants concerning one of the main problems of Coding Theory - estimating the efficiency of a code in terms of errors correction - has not yet been properly explored In this work, we consider a q-ary erasure channel and give an expression for the error probability Proposition 22 that separates the variables of the problem namely, the code and the channel, where for error probability we mean either ambiguity probability or the decoding error s probability Considering the hierarchy and spectra of generalized weights, we are able to get new bounds for the error probability of linear codes Theorem 44 It turns out that, in many cases, the upper bounds for ambiguity are better then the ones determined by Didier in [12] and the lower bound better then those determined by Fashandi et al in [13], where the authors are concerned mainly with codes over large alphabets In the last section V we consider separation properties MDS and generalizations, we show that for MDS and AMDS codes the upper and lower bounds obtained for error probability collapse, becoming hence an exact expression for the error probability, what was already known to Fashandi et al [13] in the MDS case and conclude by showing the minimizing role of MDS and AMDS codes when considering and overall error probability sufficiently small A Erasure Channel II BASIC DEFINITIONS AND NOTATION In this work we consider a Discrete Erasure Channels DEC defined by an input alphabet X = F q finite field with q elements, an output alphabet Y = F q {ɛ} where ɛ / F q is called the erasure symbol and a probability function P i j := Pr [Y = j X = i] defined by: a P j i =, for i j and {i, j} X ; b P i i = 1 p, for π 1; c P ɛ i = p, for i X, The constant < p < 1/2 is called the overall error probability of the channel A Discrete Memoryless Erasure Channels DMEC is obtained by defining n P y x = P yl x l, 1 l=1 where, x = x 1, x 2,, x n X n and y = y 1, y 2,, y n Y n B Generalized weights Given integers r, s Z with r < s, we denote r, s := {r, r + 1,, s 1, s} For simplicity, we write n := 1, n Let C F n q be an [n, k] q -linear code Since in this work we will consider only linear codes, this adjective may be omitted Given x F n q, the support of x is suppx = {i n ; with x i } and the Hamming distance may be expressed as dx, x = suppx x, where A denotes the cardinality of A Given a subcode D C, the support of D is defined as suppd = suppx x D and the generalized weight C of D is defined as C = min{ suppd ; D C and dim D = i}, for i k It is well known Wei s Theorem [1] that the generalized weights are strictly increasing d 1 C < d 2 C < < d k C and we call {d 1 C, d 2 C,, d k C} the weight hierarchy of C We denote by A i j = Ai j C, i k and j n, the set of all i-dimensional linear subcodes D C supported by j coordinates, that is, A i jc = {D C; suppd = j and dim D = i}

2 The cardinality of A i j is called the i-th generalized spectra with support j of the code C and we denote A i j = A i j We call the matrix A i j the code C Ambiguity,,k;j=, n the spectra-matrix of Considering a code C F n q and a DMEC, some messages in Y n will never be received We denote by E C the subset of messages that may be received, that is, E C = {y Y n ; P receive y } and call it the set of admissible or possible messages, where P receive y is the probability to receive y Y n Given x X n, we denote by P send x the priori probability of x so that n E C = {y Y n ; x C with P send x P yl x l } Given y Y n let R := Ry = {i n ; y i = ɛ} and define the set [y] R of R-ambiguities of y as l=1 [y] R := {x C; P y x } We note that [y] R depends both on y and C We remark that, when using a Maximum Likelihood decoder, once the message y is received, the elements of [y] R are the possible choices for decoding y We say that y is an ambiguity of C if [y] R > 1 We identify F n q with the product Π n i=1 F q i and, given R n, we denote by π R the projection of x = x i i n in the coordinates of R: π R x = x i i R To shorten the notation we will write π R x = x R We also denote by E R the set of admissible messages that has an erasure on the coordinates in R, that is, E R = {y E C ; y i = ɛ, i R} Given R n let R := {i n ; i / R} be the complement of R The following proposition consists of a sequence of elementary properties that are stated without a proof for future reference Proposition 21: Considering a DMEC, y Y n, R n and C a linear code, we have that i [y] R = iff y / E C ; i [y] R = {x C; x R = y R}; iii [] R is the kernel of the projection map π R restricted to the code C; iv For any y E C, [y] R = [ R ] ; v E R = q k 1 D Error probability for ambiguity and decoding We are considering an DMEC with conditional probabilities defined by 1, with overall error probability p and prior probabilities identically distributed, that is, P i = q 1, i F q Given y Y n we denote by P receive y the probability that y is received and by P amb y the probability that y is ambiguous The ambiguity probability of an [n, k] q -code C or the error probability before decoding procedures is P amb C = P amb yp receive y y Y n = P amb yp receive y, y E C where the last equality follows from statement i in Proposition 21 Considering a maximum likelihood decoding criteria, we denote by P dec y the probability of y Y n being decoded incorrectly and define the decoding error probability of an [n, k] q -code C as P dec C = y Y n P dec yp receive y = y E C P dec yp receive y, where the last equality again follows from statement i in Proposition 21 We will use P C to denote either P amb C or P dec C, that is, we may consider to mean either dec or amb, so that both the previous expressions may be written as P C = y E C P yp receive y 2 Since we are assuming that P i = q 1, i F q, the probability that a message y is receives P receive y = x C P y xp send x = 1 q k x C P y x Considering an admissible message y E C, we have P receive y = x [] R q k p R 1 p R 3 = q k p R 1 p R 4 Substituting expression 3 into equation 2 we get P C = y E C P yq k p R 1 p R, 5 hence we may write 5 as P C = P y E H q k p R 1 p R R n

3 and, from statement v in Proposition 21 it follows that P C = P Rp R 1 p R 6 R n We note that we are exchanging P y by P R and this is possible since those probabilities do not depend on y but only at what are the erased coordinates of y, that is, on the set Denoting we may write 6 as We note that R = Ry = {i n ; y i = ɛ} Q,r = P C = Q dec,r = Q amb,r = R; R =r P R, 7 n Q,r p r 1 p n r 8 r= {R n ; R =r} {R n ; R =r} where α is the smallest integer greater or equal to α R In our case, we have that 1 1 { if []R = 1; = 1 if > 1, that is, it equals 1 or according if there is more then one or only one namely y admissible messages having R as a set of ambiguous coordinates We define a i r := a i rc = {R n ; R = r and dim[] R = i} 11 and { 1 1 q, for = dec; δ,i = 1 i 1q, for = amb i We note that each a i r depends on the code C, since [] R is the kernel of the projection π R restricted to C We also remark that k n a i r = 12 r With this notation we have that k Q,r = a i rδ,i 13 so that equation 6 may be expressen a vectorial form: Proposition 22: The ambiguity probability and the decoding error probability of a linear code may be expressed as the product P C = δ Λρ T 14 where δ = δ,,, δ,k, ρ = 1 p n, p1 p n 1,, p n, ρ T is the transpose of the vector ρ and a a 1 a n Λ = 15 a k a k 1 a k n We remark that δ depends only on the the parameters [n, k] q and on meaning decoding or ambiguity ; ρ depends only at the channel not on the code; the matrix Λ depends only on the code C, not on the channel neither on meaning decoding or ambiguity For this reason, since only Λ depends on the code C, when looking for bounds, we will focus the attention on this matrix, which we call the supportmatrix of C III SUPPORT-MATRIX AND SPECTRA-MATRIX OF A CODE In this section and from here on, we will assume that d k = n We wish to establish a relation between the support-matrix and the spectra-matrix of a code and we start with two simple lemmas Lemma 31: Let C be a code, D A i rc and s n Then, D contains a set of linearly independent vectors {x 1,, x i 1 } D such that π s x j =, for any j i 1 Proof: Let R = suppd and let us assume first that s / R But this implies that π {s} x =, for every x D and the result follows from the fact that dimd = i Let us now assume that s R Since R = suppd, there is x D such that π {s} x We consider a basis of the subspace D containing x, let us say {x, u 1, u i 1 } We define x j = u j π su j π s x x ant is immediate to check that {x 1,, x i 1 } is linearly independent and π s x j =, for any j i 1 Lemma 32: For every i k the coefficient A i of the spectra-matrix depends on the number of different supports attained by subcodes in A i, that is, A i = {R; R = supp D and D A i } Proof: It is enough to prove that for D 1, D 2 A i, if D 1 D 2, then supp D 1 supp D 2 For D 1 D 2 we may assume wlog there is x D 2 \ D 1 We suppose suppd 1 = suppd 2 = R and will show that this leads to a contradiction Since x D 2 \ D 1, the subspace D = {x} D 1 has dimension i + 1 and since suppx suppd 1 = R, we have that suppd = R and, from Lemma 31, given s R there is a linearly independent set {x 1,, x i } D such that π {s} x j =, j i Denoting D = x 1,, x i we have that dim D = i and supp D R \ {s}, that is, supp D <, a contradiction We are now able to determine a i r = for r :

4 Proposition 33: For any i = 1, k, the coefficients of the support-matrix of a code C satisfy { for r < a i di ; r = A i for r = Proof: We first consider the case r = If we put B = {R n ; dim [] R = i and R = } we have, by definition, that B = a i It follows from Lemma 32 that we only need to prove that B = {R; R = supp D and D A i } Let us consider R n with dim[] R = i and R = and let us prove that supp[] R = R We know that supp[] R R and we can not have supp[] R > R Since dim[] H = i and supp[] R R =, it follows that supp[] R = R hence a i = A i We consider now the case r < Given r <, suppose that a i r > This implies there is R n with R = r and dim[] R = i But supp[] R R and this implies r = R supp[] R, a contradiction It follows that a i r =, for all r < We remark that the condition r = is strictly necessary in Proposition 33 Considering for example the [3, 2] 2 -code generated by the vectors {1,,,, 1, 1} for r = 2 and d 2 = 3 we have that A 1 2 = 1 and a 1 2 = 3 Wei s Monotonicity Theorem states that i < j implies < d j, so Proposition 33 ensures the following: Corollary 34: If j < i then a i d j = We continue with some results that will be used to produce the expected bounds for the error probability Lemma 35: Let R n, with R = r an = dim[] R Then, for any j n \ R we have that dim[] R {j} maxk + r + 1 n, i 16 Proof: Denoting S j = R {j}, from item iii in Proposition 21 we have [] Sj = kerπ Sj Considering that may be expressed as the composition π Sj of the projections π Sj = π {j} π R 17 π π R C F n r {j} q F n r 1 q s n c se s s R c se s s S j c s e s, the classical Kernel Theorem ensures that that is, dimc = dimimπ Sj + dimkerπ Sj, But equation 17 implies dim [] Sj = k dimimπ Sj 18 dimimπ Sj mindimimπ R, dimimπ {j} 19 Since π {j} determines a projection of an n r-dimensional space into an n r 1-dimensional one, we have that dimimπ {j} = n r 1 2 and since dim[] R = dimkerπ R, we have that dimimπ R = dimc dimkerπ R = k i 21 It follows from 19, 2 and 21 that dimimπ Sj mink i, n r 1 22 and equations 18 and 22 together imply dim[] Sj k mink i, n r 1 = maxi, k + r + 1 n The next Propositions will be usen the proof of our new bounds in Theorem 44 and follows easily from Lemma 35 Corollary 36: Given an [n, k] q -linear code C and R n with R = r, then q k n+r Proof: The proof is made by induction on R For the initial step, R =, the result is satisfied since R = Suppose q k n+ R for every R n with R r and let us prove it also holds for J n with J = r + 1 We write J = R {j} with R = r and j / R, and from Lemma 35 it follows that dim[] J = dim[] R {j} maxk + r + 1 n, i with i = dim[] R The induction hypothesis implies that i k n + r hence dim[] J maxk + r + 1 n, k n + r = k n + r + 1 Proposition 37: If r n k + i + 1, then a i r = Proof: Suppose a i r > for some r n k + i + 1, that is, suppose there is a set R n with dim[] R = i and R := r n k + i We cannot have R = n, since this woulmply r = n and i = k, contradicting inequality 23 So, let us assume that R n, so there is j n \ R From Lemma 35 we have that dim[] R {j} maxk + r + 1 n, i and from inequality 23 we have k + r + 1 n i + 2 > i, hence dim[] R {j} i + 2 It follows there is a subcode D [] R {j} with dimd = i+2 and Lemma 31 ensures the existence of a subcode D D such that dim D = i + 1 and supp D R But D []R and dim[] R = i, a contradiction It follows that, for r n k+i+1, there does not exist R n such that R = r and dim[] R = i, in other words, a i r = for r n k + i + 1

5 Corollary 38: a i n 1 = for every i k 1 and a k 1 n 1 = n Proof: Considering r = n 1 = n k + k 1 and i < k 1, Proposition 37 ensures a i n 1 = Proposition 33 it follows that a k n 1 = a k d k 1 = and as a particular case of expression 12 we have that k n a i n 1 =, n 1 and, since a k 1 n 1 is the only non-zero summann the equality above, we have a k 1 n 1 = n n 1 = n IV BOUNDS FOR P In this section, we establish bounds for P by founding bounds for the coefficients Q,r definen equality 13 We start with three lemmas that give us values and bounds for Lemma 41: Let C be an [n, k] q -linear code and let D C be an i-dimensional linear subcode of C If suppd R n, then q i Proof: Since D [] R, we have that D = q i Lemma 42: Let C be an [n, k] q -linear code If R = suppd and D A i, then = q i Proof: From item iii in Proposition 21 we know that [] R is a vector subspace of F n q and hence is a power of q We assume that q i+1 and this will lead us to a contradiction But q i+1 implies dim[] R i + 1 hence there is a subspace D [] R with dimd = i + 1 From D [] R it follows that hence suppd supp[] R R +1 suppd supp[] R H = But this contradicts the fact < +1, ensured by the Monotonicity Theorem Section II-B So we have that q i and Lemma 41 ensures = q i In the previous lemma we considered a subset R n that is the support of a subcode D A i realizing the i-th weight In the following proposition we assume that R = but R suppd for any D realizing the i-th weight Lemma 43: Let C be an [n, k] q -linear code If R n satisfies R = but R suppd for any D A i, then q i 1 Proof: Suppose q i, or equivalently, dim[] R i In this case, there is an i-dimensional D [] R of C such that suppd R and suppd supp[] R R =, where the first inequality follows from the minimality of Then we have suppd = R, dimd = i and suppd =, contradicting the hypothesis that R suppd for any D A i So, < q i and we get q i 1 Now we are able to establish bounds for P C This will be done in the next theorem, that actually establish upper and lower bounds for some of the coefficients Q,j in expression 8 Theorem 44 Bounds for P C: Let C be an [n, k] q -linear code Then, a For every i k, the coefficient Q dec,di is greater or equal to A i 1 1 q + n i A i 1 di 1 ; max{1,q k n+ } b For every i k, the coefficient Q dec,di is smaller or equal to q 1 n A i q i q i 1 ; c For every i 2, k, the coefficient Q amb,di is greater or equal to n A i 1 + A idi 1 max{1, q k n+di } d Q amb,d1 = A 1 d 1 Proof: a To simplify the notation we write: and Φ i = {R n ; R = suppd with D A i } Φ i = {R n ; R = } \ Φ i Using this notation and expression 9, the coefficient Q dec,di is expressed as Q dec,di = R Φ i 1 1 [] R R Φi [] R 24 Lemma 42 ensures that R Φ i implies = q i so Q dec,di R Φ i 1 1 q i + R Φ i 1 1 [] R Corollary 36 ensures that if R Φ i then q k n+di and since we must have 1 1 for it represents a probability, we get that Q dec,di R Φ i 1 1 q + 1 i R Φ i 1 max{1,q k n+ } From Lemma 32 we have that Φ i = A i and since the summands do not depend on R we get that Q dec,di is greater or equal to A i 1 1 q + n i A i 1 di 1 max{1,q k n+ } b From Lemma 43 we can bound q i 1, for R Φ i, and from Lemma 42 we obtain an expression to, for R Φ i Substituting those values in expression 24 we get that Q dec,di A i q 1 q i + n 1 1 q i 1 c From Lemma 42 we have 1 1 = 1 25

6 and, from Corollary 36, max{1, q k n+di } We write the expression 1 as Q dec,di = R Φ i 1 1 [] R R Φi [] R, 27 an substituting it into 25 and 26 we get Q amb,di A i + n A i 1 di 1 28 max{1,q k n+ } d From expression 13 we have that Q amb,d1 = k a i d 1 1 q i From Corollary 34, only two of the summands above are non zero, namely Q amb,d1 = a d 1 1 q + a 1 d 1 1 q 1 = a 1 d 1 1 q 1 and from Proposition 33 we have Q amb,d1 = a 1 d 1 = A 1 d 1 V P AND SEPARABILITY PROPERTIES We start this section presenting some separability properties that generalize the concept of MDS codes and then we will study the behavior of the bounds for P C expressen Theorem 44 for codes having some of those separability properties Using the Singleton bound d 1 C n k + 1 we may define the Singleton defect sc of an [n, k] q -code C as sc = n k + 1 d 1 C Using the defect, we say that a code C is Maximum Distance Separable MDS if sc = and following Boer in [14] C is Almost Maximum Distance Separable MDS if sc = 1 Considering the generalized weights, the separability property of a linear code may be expresses in different but natural ways Considering the monotonicity of the weight hierarchy, the i-th Singleton defect of an [n, k] q -linear code C is defined as s i C = n k + i C We say following Wei in [1] that C is an j-mds code if s j C = and j-amds code if s j C = 1 We say that C is a proper j-mds code or just P j -MDS if it is j-mds and proper in the sense that j = min{i k ; C is an i-mds code} Similarly, we say C is an P j -AMDS code if j = min{i k ; C is an i-amds code} A Expressions for P amb C and P dec C We consider the matrix Λ usen the vectorial form used in Proposition 22 to express the ambiguity or the decoding error probability Propositions 33 and 37 ensures many of the coefficients of Λ T are null Let us write Λ T explicitly as: 1 n d 1 1 n d 1 A 1 d1 A 1 d 1 a d 1+1 a 1 d 1+1 a d k 1 1 a 1 d k 1 1 a d k 1 a 1 d k 1 A k 1 d k 1 Λ T = a d k 1 +1 a 1 d k 1 +1 ak 1 d k 1 +1 a n k a 1 n k a k 1 n k a 1 n k+1 ak 1 n k+1 a k 1 n k+2 a k 1 n 2 n A k d k In this presentation, the blue values are ensured by Proposition 33 and the green entries by Proposition 37 Looking at expression 12, we see it is summing over the lines of Λ T so, in the lines where at most one entry is unknown, we can determine the remaining one using 12: those are the three entries in red Looking now at the columns of Λ T, we see that the quantity of undetermined entries at the column j is given by the difference n k+i and, in an informal way, we can state that the more C is separable, the more the entries of Λ T are known In particular, assuming that C is MDS, that is, that d 1 = n k + 1, the monotonicity of the weights implies that = n k + i for every i k ann this case, all nonzero entries of Λ are expressen terms of the weight spectra, namely, in terms of A 1 d 1, A 2 d 2,, A k d k But for an MDS code, the following theorem due to Han, in [11] gives explicit expressions for those coefficients, depending exclusively on n,k and q: Theorem 51 Theorem 25 in [11]: Let C be an [n, k] q - linear code and suppose that C is P s -MDS Then, for s i k, we have that { A i, if r rc = n r di r t= 1t [ r r+i di t] t i if < r n Using those expressions for A 1 d 1, A 2 d 2,, A k d k we have an alternative proof of the following Theorem already proved by Kasami and Lin in [15]: q,

7 Theorem 52: Let C be an MDS code Then, a P amb C = n i=n k+1 i p i 1 p n i ; b P dec C = n i=n k+1 i 1 1 q p i 1 p i i If C is an AMDS code, that is, if d 1 = n k, there is an unique s := sc k such that C is P s -MDS and we can determine an explicit formula for P C depending only on A 1 d 1 and s: Theorem 53: Let C be an [n, k] q AMDS linear code and let s := sc k such that C is P s -MDS Then, and P amb C =A 1 n kp n k 1 p k n n + p n k+i 1 p k i n k + i s 2 P dec C = + k i=1 A i+1 n k+i n n k + i q 1 q i+1 p n k+i 1 p k i 1 1q i p n k+i 1 p k i Proof: It follows straightforward from the use of the vectorial form 22 and Propositions 33 and 37 We remark that Theorem 53 ensures that, for an AMDS code, the error probability P amb C and P dec C of a code is completely determined by the coefficients A 1 n k and A 1 n k,, As 1 n k+s 2 of the spectra-matrix, respectively It follows that bounds for the coefficients of the spectra-matrix leads to bounds for P C In the particular case of an NMDS-code a code C such that d 1 C = n k and d 2 C = n k + 2, the coefficient A 1 n k fully determines P C and an upper bound for this coefficient is provided Dodunekov e Landgev, in [16]: A 1 n k n q 1 k 1 k B Behavior of P for small p and optimality of MDS and AMDS codes As expected, for small overall error probability p, minimizing error probability P C demands to maximize d 1 C: Proposition 54: Let C 1 and C 2 be two [n, k] q -linear codes For p sufficiently small, if d 1 C 1 > d 1 C 2 then P C 1 < P C 2 Proof: To prove the proposition we assume d 1 C 1 > d 1 C 2 and show that P C 1 p P C 2 = But P C 1 p P C 2 = p = p i=d 1C 1 Q,iC 1 p i 1 p n i j=d 1C 2 Q,jC 2 p j 1 p n j i=d 1C 1 Q,iC 1 p 1 p j=d 1C 2 Q,jC 2 p 1 p i j, where the first equality follows from equation 8 Denoting x = p 1 p and noting that p p 1 p = we get that P C 1 x P C 2 = i=d Q 1C 1,iC 1 x i x j=d Q 1C 2,jC 2 x j d1c 2 i=d = Q 1C 1 d 1C 2,i+d 1C 2C 1 x i x d1c 2 j= Q,j+d1C 2C 2 x j and since we are assuming d 1 C 1 d 1 C 2 >, we have that P C 1 x P C 2 = < 1 and from continuity of the functions we have that P C1 P < 1, C 2 or equivalently P C 1 < P C 2, for every p sufficiently small If for a given pair n, k there exist an MDS AMDS [n, k] q code, we say that n, k is q-mds q-amds As an immediate consequence of Proposition 54 we have the following proposition already known and proven [13]: Proposition 55: If n, k is q-mds and C is an [n, k] q -code that minimizes the error probability, then C is MDS Pairs that are q-mds are not very frequent For q = 2, for example, it is well known that MDS-codes are rather trivial and the unique 2-MDS pairs are the pairs n, n, n, 1, n,, n, n 1 Considering AMDS codes, those are not classified, but there are many constructions of particular families of AMDS codes and results ensuring the existences of such codes with parameters n and k see for example [14] In all those cases, when the pair n, k is AMDS but not MDS, for p sufficiently small, a code that minimizes P should be an AMDS code From Proposition 54 we know that, for p sufficiently small, we should look for codes having maximal minimal distance Among all those codes, which should perform better? A partial answer is given by the next two results and can be summarized as follows: maximize the minimal distance and then minimize the corresponding value in the spectra Proposition 56: Let C 1 and C 2 be two [n, k] q -linear codes with d 1 C 1 = d 1 C 2 If Q,d1C 1 < Q,d1C 2, then P C 1 < P C 2, for p sufficiently small Proof: From equation 8 we get that i=d Q 1C 1,iC 1 x i i=d Q 1C 2,iC 2 x i P C 1 x P C 2 = x We write d 1 = d 1 C 1 = d 1 C 2 and cancel x d from the right side we get P C 1 x P C 2 = Q,d1 C 1 + i=d Q 1+1,iC 1 x i d1 x Q,d1 C 2 + i=d Q 1+1,iC 2 x i d1 = Q,d 1 C 1 Q,d1 C 2 < 1 hence P C 1 < P C 2 for every p sufficiently small Proposition 57: Let C 1 and C 2 be two [n, k] q -linear codes with d 1 C 1 = d 1 C 2 If A 1 d 1 C 1 < A 1 d 1 C 2, then P C 1 < P C 2, for p sufficiently small Proof: It follows straightforward from Proposition 56 and items b and d of Theorem 44

8 VI CONCLUSION In this work we used the generalized weights and spectra to set new bounds for the error probability over an erasure channel Further work may be done exploring the situation when two codes have the same minimal distance and this is attained by the same number of vectors The role of generalized weights and spectra for the error probability still needs to be explained for other channels ACKNOWLEDGMENT This work was partially supported by CAPES and FAPESP grants 27/ and 213/9493- REFERENCES [1] V K Wei, Generalized hamming weights for linear codes, Information Theory, IEEE Transactions on, vol 37, no 5, pp , 1991 [2] H Stichtenoth and C Voss, Generalized hamming weights of trace codes, Information Theory, IEEE Transactions on, vol 4, no 2, pp , 1994 [3] G Feng, K Tzeng, and V Wei, On the generalized hamming weights of several classes of cyclic codes, Information Theory, IEEE Transactions on, vol 38, no 3, pp , 1992 [4] C Munuera, On the generalized hamming weights of geometric goppa codes, Information Theory, IEEE Transactions on, vol 4, no 6, pp , 1994 [5] A Barbero and C Munuera, The weight hierarchy of hermitian codes, SIAM Journal on Discrete Mathematics, vol 13, no 1, pp 79 14, 2 [Online] Available: [6] H Georg Schaathun, The weight hierarchy of product codes, Information Theory, IEEE Transactions on, vol 46, no 7, pp , 2 [7] P Heijnen and R Pellikaan, Generalized hamming weights of q-ary reed-muller codes, Information Theory, IEEE Transactions on, vol 44, no 1, pp , 1998 [8] T Klove, Weight hierarchies of binary linear codes of dimension 4, in Information Theory, 1993 Proceedings 1993 IEEE International Symposium on, 1993, pp [9] J Hirschfeld, M Tsfasman, and S Vladut, The weight hierarchy of higher dimensional hermitian codes, Information Theory, IEEE Transactions on, vol 4, no 1, pp , 1994 [1] A Ashikhmin, A Barg, and S Litsyn, New upper bounds on generalized weights, Information Theory, IEEE Transactions on, vol 45, no 4, pp , 1999 [11] S T D S Han, Higher weights and generalized mds codes, J Korean Math Soc, vol 47, no 6, pp , 21 [12] F Didier, A new upper bound on the block error probability after decoding over the erasure channel, Information Theory, IEEE Transactions on, vol 52, no 1, pp , 26 [13] S Fashandi, S Gharan, and A Khandani, Coding over an erasure channel with a large alphabet size, in Information Theory, 28 ISIT 28 IEEE International Symposium on, 28, pp [14] M A de Boer, Almost mds codes, Designs, Codes and Cryptography, vol 9, no 2, pp , 1996 [15] T Kasami and S Lin, On the probability of undetected error for the maximum distance separable codes, Communications, IEEE Transactions on, vol 32, no 9, pp , 1984 [16] S Dodunekov and I Landgev, On near-mds codes, Journal of Geometry, vol 54, no 1-2, pp 3 43, 1995 [Online] Available: