THE development of efficient and simplified bounding

Transcription

1 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 ist ermutation invariant linear codes: Theory and Alications Kostis Xenoulis Abstract The class of q-ary list ermutation invariant linear codes is introduced in the current work along with robabilistic arguments that validate their existence when certain conditions are met. The secific class of codes is characterized by an uer bound that is tighter than the generalized Shulman-Feder bound and relies on the distance of the codes weight distribution to the binomial multinomial resectively one. The bound alies to cases where a code from the roosed class is transmitted over a q-ary outut symmetric discrete memoryless channel and list decoding with fixed list size is erformed at the outut. In the binary case, the new uer bounding technique allows the discovery of list ermutation invariant codes whose uer bound coincides with shere-acking exonent. Furthermore, the roosed technique motivates the introduction of a new class of uer bounds for general q-ary linear codes whose members are at least as tight as the DS bound as well as all its variations for the discrete channels treated in the current work. Index Terms Discrete symmetric channels, double exonential function, list decoding, ermutation invariance, reliability function. I. ITRODUCTIO THE develoment of efficient and simlified bounding techniques which aly to structured families of codes constitutes a maor research area in the field of coded communications. Their imortance lies in their ability to designate characteristics under which the roosed codes achieve low error decoding robability with rates close to the caacity of the considered channels. A vast maority of bounding techniques have been constructed for structured ensembles of codes over discrete and continuous channels, as it is reorted in the literature []. Their usefulness relies among others on their ability to aly to secific families as well as to fully random ensembles of codes. A classical reresentative of the class of these techniques, which alies to both categories of codes, is the generalized second version of the Duman-Salehi bound DS [, Section II.B]. For secific codes, many reorted bounds such as Shulman-Feder bound SF [3] for discrete memoryless channels and Divsalar bound [4] for binary inut additive Gaussian channels, consist secial cases of the DS bound. The latter can be alied to the fully random ensemble of codes and rovide Gallager s random coding theorem [5]. An interesting class of bounding techniques for the error decoding robability of secific codes is the one which enoys random coding characteristics, such as the SF bound and its This research has been co-financed by the Euroean Union Euroean Social Fund-ESF and Greek national funds through the Oerational Program Education and ifelong earning of the ational Strategic Reference Framework SRF - Research Funding Program: Aristeia I-5. The author is with the Deartment of Informatics and Telecommunications, ational and Kaodistrian University of Athens, Athens, Greece 5784, e- mail: kxen@di.uoa.gr generalized version, reorted in [, Aendix A]. The latter relates the coding exonent of nonrandom codes to the exonent deduced from the random coding analysis of Gallager [5]. Secifically, through the construction of an artificial ensemble of codes from a single block code, the difference between the error exonent of the secific code and the exonent of the random coding ensemble under maximum likelihood M decoding is revealed to be equal to a constant term. This term deends only on the weight sectrum of the code and its relation to the binomial multinomial distribution and not on the channel s characteristics. The imortance of the resented bounds lies on their close similarity to the shere acking lower bound [6] on the error decoding robability and its tightest versions for finite blocklengths, resented in [7] and [8] resectively. The urose of the current work is to rovide new tight uer bounds for the fixed size list error decoding robability, both in the sirit of the SF bound and its generalized version, that are able to aroach the shere-acking lower bound. Exonential error bounds and random coding exonents for list error decoding with variable list size have been established in [9] for the class of randomly constructed codes. Similar results under fixed-size list decoding were resently recently in [] for the class of fixed-comosition codes. Regarding secific codes, error erformance bounds under fixed-size listdecoding are rovided in [] and []. In [] the bounds secialize to linear codes and continuous outut channels and rely on the distances between the codewords on the list. On the other hand, [] focuses on the derivation of Gallagertye bounds both under variable and fixed size list decoding. The resented bounds are message indeendent for generalsymmetric channels and aly to both random and structured linear block codes or code ensembles. Regarding the case of structured code ensembles, the bounds resented in the current work are deloyed by relacing weighted sums of likelihood ratios in the DS technique as well as in the roof of the random coding theorem by double exonential functionals. We erform so in order to exloit the concave behavior of the double exonential function in the error decoding region of a code under fixed size list decoding and combine it with the random coding technique. To roceed further, the class of -list ermutation invariant linear codes is introduced. It is revealed that every member of the class enoys the same list error decoding robability for the transmission of coded information over discrete memoryless symmetric channels. The secific aroach constitutes a new design aradigm where the need for bounding techniques, that lead to tight closed-form uer bounds, enables the definition of a new class of codes. For a list ermutation invariant linear

2 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 code, the roosed bounding technique leads to tight uer bounds in relation to SF bound and its generalized version while it still relies on the similarity of the code s weight distribution to the binomial one. The analytic construction of members belonging to the revious class of codes constitutes a heavy comutational rocess since it relies on the satisfaction of certain roerties of a code s distance distribution matrix. evertheless, the robabilistic method [3] and secifically ovasz s ocal emma allows the roof of existence of such a class of codes and renders them as a useful tool in the calculation of the reliability function of discrete memoryless channels for certain coding rates. The double exonential technique, as described in the revious aragrah, is relative new and has aeared in [4]. In general, double exonential bounds have aeared in the analysis of error exonents for channels with feedback, such as in [5] and references therein, but are not related to the resented technique. The latter can also be alied to general q-ary linear codes leading to a tighter version of the DS bound. evertheless, it violates the need for simlified bounds since it requires the knowledge of fine details of a code s characteristics, such as the full coset weight distribution. The revious obstacle is circumvented by noting that the key ingredient of the DS bound is a weighted average of the sum of likelihood ratios of the code s codewords with resect to the transmitted one, and that this sum can be relaced with resective sums of horizontally fli sigmoid functions. The latter manage to smooth the effect of the erroneous decoding region in the calculation of the list error decoding robability and thus achieve a tradeoff between the desirable characteristics of bounding techniques. amely, the resented bounds are tighter than the DS bound while their evaluation does not require any extra knowledge in comarison to the DS bound. The resent aer extends the work in [4] with the introduction of the class of q-ary -list ermutation invariant linear codes and the analysis of the error erformance of its members under list decoding for transmission over q- ary discrete symmetric memoryless channels. Towards the extension of list ermutation invariant codes from binary to q-ary fields and in comarison to [4], a new definition of - list ermutation invariant codes is rovided in order to address a broader class of candidate codes. Even though the extension to the general q-ary case is straightforward, the derivation of an analytic error bound exression is rather comlicated. In what follows, Section II secifies the roerties underinning the class of -list ermutation invariant linear codes and resents a relaxed condition under which such codes aear to exist. Under this condition, subsection II-A validates the existence of ermutation invariant codes and enables the analysis of their list-error decoding robability, leading to the uer bounds resented in Sections III and III-A resectively. The class of ermutation invariant codes is utilized in Section IV to rovide the reliability function of binary symmetric channels for certain coding rates. Finally, a new class of bounding techniques is resented in Section V for general q-ary linear codes, while Section VI concludes the analysis by numerically comaring the develoed bounds for the new class of codes. II. PERMUTATIO IVARIACE et the transmission of an arbitrary set of messages M, with cardinality M, over a discrete memoryless q-ary outut symmetric channel, with the use of an, R linear code C defined over the finite field F q,,..., q }. Secifically, each message m [, M ] with M = q R, is encoded onto an length codeword c m Fq of C, where c is the all-zero codeword, and is transmitted over the symmetric channel described by the transition robability Pry c m, where y Fq the length sequence at the outut of the channel. During the transmission, each symbol of a codeword remains the same with robability, while it is changed to another of the remaining q symbols with robability /q. In the analysis that follows, we denote by d min the minimum Hamming weight of the linear code C and by S l,...,l q the number of codewords of C having a fixed tye [6, Ch. ] with l i occurrences of symbol i. Furthermore, we denote the weight distribution of a linear code C over F q as the sequence S k } k=d min where S k S l,...,l q. l +...+l q =k The translate set y C y c, c C} is called a coset of the linear code C, while the vector with the smallest Hamming weight in the aforementioned set is called a coset leader. The weights of the vectors in the coset y C define the weight distribution of the considered coset, while the largest weight of a coset leader equals the covering radius of C, denoted by C. The relative radius of C is defined resectively as r C C/. Definition : The distance distribution matrix Γ of a linear code C is defined as the K + matrix having as rows all the ossible K different coset weight distributions. The first row of Γ defines the weight distribution of C, while all other rows are identified by the weight of the coset leader coset weight. If the weight distribution of the coset of the received vector y belongs to the i-th row of the matrix, then y has t k= Γ i,k codewords within Hamming distance t. Furthermore, the -th column of Γ, [, ], denotes the number of codewords a vector y of Hamming weight i can have at exact Hamming distance. Considering list decoding is erformed at the outut of the channel with list size, the conditional error decoding robability given the transmission of message is Pe,C = Pry c, C y Y,C where Y,C y Fq : l i } i= } with Pry c li, C Pry c, C, l i. 3 Since the channel is memoryless and outut symmetric, the error decoding region Y,C is equivalently exressed as Y,C = y Fq : l i } i= } with wty c li wty, l i 4

3 XEOUIS: IST PERMUTATIO IVARIAT IEAR CODES 3 where wty denotes the Hamming weight of a q-ary vector y, with the resective algebra defined over the finite field F q. Moreover, if for λ,, we define as in [4] Ωy, C, λ, then, for every y Y,C it holds Pry c λ m Pry c m λ 5 f de Ωy, C, λ,, f de x ex ex x 6 and the error decoding robability Pe,C in is uer bounded for any λ, as Pe,C Pry c, Cf de Ωy, C, λ,. 7 y Y,C It is noted that Ωy, C, λ, is exactly the inverse of the exression used to bound the indicator function φ m y in [5, Eqs. 4,5]. If we roceed by extending the above sum over the set Fq, then we cannot easily exloit the concavity of the double exonential function. Thus we cannot average over the random ensemble of codes and emloy Jensen s inequality to roduce a random coding exonent. This is due to the fact that f de Ωy, C, λ, is concave with resect to Ωy, C, λ, only for y Y,C. To surass the revious obstacle and take advantage of the secial form of 7, we seek for a family of linear codes where each code has the same error decoding region Y,C. This motivates us to define the class of -list ermutation invariant linear codes. Definition : An, R linear code C with distance distribution matrix Γ is called -list ermutation invariant if both the following roerties are satisfied: : there exists a ositive integer w ot [ d min /, / such that and min w ot κ [,K] w= Γ κ,wot + 8 max w ot Γ k,w. 9 κ [,K] w= : For all κ [, K] there exists a wκ > w ot such that Γ κ,wot+ =... = Γ κ,w κ = and Γ κ,w κ +. In case =, we call the secific code M ermutation invariant. Examle : As an examle, we consider the ternary [3, 7, 5] quadratic-residue code of [7] with distance distribution matrix deicted in Table I. If we select w ot = d min / = 3, then for =, roerty is satisfied, since the sum of the columns before the 3-rd one is lower than or equal to. Furthermore, roerty is also satisfied since for every row, in columns 4 and 5 there is an element greater than. Consequently, the secific code is M ermutation invariant. With similar arguments, selecting w ot = 4 let us deduce that the secific code is also 6-list or 7-list or 8-list or 9-list or -list ermutation invariant. Relation max κ [,K],w<wot Γ k,w was used in [4, Def. ] instead of 9. TABE I DISTACE DISTRIBUTIO MATRIX Γ OF THE TERARY [3, 7, 5] QUADRATIC-RESIDUE CODE OF [7, TABE I] Remark : Proerty in fact guarantees that if a vector y of Hamming weight w ot d min /, which causes an -list decoding error, is ermuted then the ermuted vector still causes an -list decoding error. amely, consider a vector y of weight w ot that has Γ κ,wot codewords at exact Hamming distance w ot. Then, y will have w ot w= Γ κ,w codewords within distance w ot. Consider now a ermuted vector y of y and suose that it has Γ κ,w ot codewords at exact distance w ot where κ κ. Then according to 8, the ermuted vector of y will still have w ot w= Γ κ,w + codewords within Hamming distance. Furthermore, the secific roerty in 9 guarantees that all vectors of Hamming weight lower than w ot do not cause an -list decoding error. Regarding roerty, the latter ensures that all vectors y of Hamming weight greater than w ot do cause an error. Indeed, under the validity of 8, in every row of the matrix Γ, every sum of the elements of the resective row u to w for every w > w ot will be greater than +. As in [4], from an -list ermutation invariant linear code C we create an ensemble of linear codes E by considering all ossible osition ermutation codes PC. A PC code is constructed by multilying every codeword of C with a osition ermutation matrix P which has a single in every row and column and is orthogonal, PP T = P T P = I, where I is the identity matrix and P T the transose matrix of P. The ermutation ensemble E is utilized in order to rove the following roerty for an -list ermutation invariant linear code. emma : For an, R linear code C that is -list ermutation invariant, all codes in the osition ermutation ensemble E have the same error decoding region Y. Proof: The roof of the lemma follows the same line as in the roof of [4, emma ]. In articular, suose that in the original -list ermutation invariant code C the received vector is y Y,C where the weight distribution of the unique coset y C is Γ ν,w } w=. Due to the restriction imosed by condition, it must be wty w ot. Indeed, if wty < w ot, then according to 9 y will have at most codewords, not including the all zeros codeword c, within Hamming distance wty. Thus this y will not cause a list decoding error.

4 4 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 et such a received vector y Y,C with wty = w ot. Then y has w ot k= Γ ν,k codewords, different from c, within Hamming distance w ot. et, for any ermutation matrix P, the corresonding ermuted vector P T y of y Y,C. Vector PT y is also of weight w ot while the weight distribution of coset P T y C is denoted by Γ ξ,w } w=, where ξ does not necessarily equal ν. Since is selected as the minimum sum over all rows of Γ u to the w ot -th column minus and w ot w= Γ κ,w for all κ [, K] roerty, vector P T y also has codewords c li } i= of C, aart from the all-zeros codeword c, within distance wty = w ot, i.e. wt P T y c li wty, i [, ]. Thus P T y Y,C. Moreover, according to the revious inequality it is noted wty wt P T y c li = wt P P T y c li = wt y Pc li. The inequality resented in indicates that y also has codewords c li } i=, aart from c, of the ermuted code PC within distance wty. Hence, y Y,C causes a list decoding error in PC. Suose now that y Y,C with weight wty = w ν w ot, wκ ], for some ν [, K], and consider the weight distribution of the corresonding coset y C, Γ ν,w } w=. For any ermutation matrix P, the weight distribution of coset P T y C is denoted by Γ ξ,w } w=, where again ξ is not necessarily equal to ν. Then, due to roerty, vector P T y has Γ ξ,w ξ codewords of C, aart from c, at Hamming distance wξ w ν. Equality holds in the latter relation in case cosets y C and P T y C have the same weight distribution. Consequently, P T y Y,C, while the weight transformation guarantees that vector y also causes a list decoding error in the ermuted code PC. Finally, in all other cases where y Y,C and wty > w κ for all κ [, K], vector P T y w κ always has min κ [,K] w= Γ κ,w codewords of C within distance wty and thus P T y Y,C. The secific roerty stated in emma constitutes the main reason for defining the class of -list ermutation invariant codes. In articular, taking the average of both sides of 7 over the ermutation ensemble E we manage to change the order of summation with resect to Y,C and E. Since the double exonential function is concave over the subset Y,C, we are able to rovide an analytic tight bound for the secific codes, as shown in Section III. An easy to track set of necessary conditions for the definition of -list ermutation invariant linear codes is rovided next according to the following lemma. emma : et for a non-negative integer t < / d min / and a ositive integer s all of the following conditions be valid for a linear code C: i every vector y Fq of weight d min / + t has at least + codewords of C within distance d min / + t. ii every vector y Fq of weight w < d min / + t has at most codewords of C within distance w. iii each vector y Fq of weight at most C has at least + codewords of C within distance d min / + t + s. Then C is -list ermutation invariant. Proof: If according to Definition we select w ot = d min / + t, then due to the first statement of the current lemma, according to which every vector y has at least + codewords within distance d min / + t, roerty of Definition will be satisfied. This is due to the fact that the minimum of the elements of the d min / + t-th column of the distance distribution matrix Γ will be at least +. Moreover, according to the second statement, the sum of all rows of Γ u to the d min / + t-th column, will be at most. Furthermore, if the third statement of the lemma is satisfied, then for every row coset weight distribution of Γ there exists a column s ositions right to the d min / + t-th one, such that each vector y with weight u to the covering radius C has at least + codewords within distance d min / +t+s. Consequently, roerty of Definition is also satisfied. Remark : For t = the second statement of emma is trivial, since all vectors y with weight w < d min / have only the zero codeword within distance w. This is the case emloyed in the analysis next. Secifically, if for t = the fixed list size satisfies > S, S C+ d min/ +s k=d min S k where terms S k are rovided by, then C is -list ermutation invariant. Indeed, according to the first statement of emma, a codeword c, aart from the all-zeros one c, which is at Hamming distance at most d min / from a vector y of Hamming weight wty = d min /, should satisfy wtc d min /. The revious statement follows due to the revious assumtions and the Hamming distance inequality according to which Therefore, if we set wtc wty d min /. S d min/ k=d min S k then every vector y of weight d min / enoys at most S codewords within distance d min /. With similar arguments, one can deduce that the Hamming weight of each codeword c satisfying the third statement of emma is constrained to wtc C + d min / + s. Thus the maximum number of codewords a vector y of weight C has within distance d min / + s is S. Since C d min / [8], it holds S < S and consequently the validity of confirms the -list ermutation invariance roerty of the linear code C. A. Existence of -list ermutation invariant linear codes The construction of general -list ermutation invariant linear codes aears to be a rather difficult rocess, since it requires the analytic calculation of the distance distribution matrix Γ of the code. In what follows, with the aid of emma

5 XEOUIS: IST PERMUTATIO IVARIAT IEAR CODES 5 and for t =, we investigate the conditions under which such codes aear to exist. Theorem : et an, R linear code C satisfy for some ositive integer s min S q Hqδ H q rc, S q δ q Hqδ} e < 3 as PrA S S y F q wty= d min dmin VolBy, d min / q q d min q Hqδ q S q δ q Hqδ 6 where δ = d min / /, δ = d min / + s/ and S defined in. If [, S ] and / > max r C, δ }, then with ositive robability C is -list ermutation invariant. Proof: Consider an, R random linear code C over the field F q constructed by choosing R linearly indeendent vectors of length over F q. et By, r denote the Hamming ball with center y and radius r, i.e. By, r = y F q : wty y r} with volume VolBy, r. If H q denotes the entroy function H q log q log q, then the volume of the considered Hamming ball is uer bounded as VolBy, d min / q Hqδ. 4 et the following arametrized events, where the all-zeros codeword c is excluded: A l = B l = y F q, c i } l i= \ c : c i } l i= fall in a Hamming } ball By, d min / with wty = d min / y F q, c i } l i= \ c : c i } l i= fall in a Hamming } ball By, d min / + s where wty C. We are interested in uer bounding the following series of robabilities PrA l, B l for every l [, ], where events A l, B l are not necessarily disoint. It is easily deduced that PrA l, B l is a decreasing function with resect to l so that we concentrate on uer bounding PrA, B. Due to inequality PrA, B min PrA, PrB } 5 we examine searately the robabilities on the right hand side of the revious inequality. In what follows, we make use of the following argument stated in [9] and in the roof of [, Theorem 5.6]. amely, every set of distinct non-zero messages in Fq k contains a subset of at least l = log q messages which are linearly indeendent over F q. It is easily verified that such linearly indeendent l-tules are maed to l mutually indeendent codewords by a random linear code. For the event A, the resective robability is uer bounded by the robability there exists a y Fq with Hamming weight wty = d min / having l = log q = codeword c out of at most S codewords within Hamming distance d min /, according to Remark. The latter robability is uer bounded according to 4 and the union bound where we have made use of the following relation n q nhq n k. k In a similar manner, the investigated robability of the event B is uer bounded as PrB S q wty qhqδ wty q y:wty C S q Hqδ H q rc 7 where the above sum with resect to wty equals the volume of a Hamming ball with center the all-zero codeword c and radius C. It is reminded that r C is the relative covering radius of the C. Combing 5-7 we have PrA, B min S q Hqδ H q rc, S q δ q Hqδ}. 8 It is noted that each event A l B l, l [, ] deends on at most others, while the robability of each event is uer bounded by the right hand side of 8. Then according to the symmetric case of ovasz ocal emma [3, Corollary 5..], if 3 is satisfied, then with ositive robability, none of the events A l B l, l [, ] are valid, and thus the statements of emma when t = are satisfied. The range of linear codes which satisfy 3 is mainly constrained by the emloyment of 5 and the use of emma for t =. The latter rovide only sufficient conditions for a linear code to be ermutation invariant. Moreover, the roof of non-trivial existence results for t > requires the develoment of new bounds in the sirit of [9]. The latter relate the number of codewords of a linear code that fall in a Hamming ball with regard to the ball s radius. Remark 3: If a linear code C is -list ermutation invariant, then it is also M ermutation invariant according to Definition. Thus, one can rove the existence of an M ermutation invariant code ust by validating 3 for a code C with =. Remark 4: The ositive integer constant s incororated in 3 through δ affects the rate of the code C for which the latter is -list ermutation invariant. Indeed, since S, S < M = q R, the sufficient condition 3 of Theorem can be relaced by the most stringent inequality min q δ q Hqδ R, q Hqδ H q rc R } e <. 9

6 6 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 Furthermore, if C is binary q =, then 9 reduces to Hδ R e < since / > min r C, δ } δ under the assumtion of Theorem. It is worth mentioning that for large is aroximated for finite by R < R H δ. Rate R is comared next with the caacity of the binary symmetric channel C = H, where denotes the error transition robability of the channel. One can easily rove by simle differentiation that H λx λh x, λ < /x, is an increasing function with resect to x when λ < and a decreasing one when λ >. If H < H δ or < δ then R < C since H δ < H δ. Remark 5: For finite list size and infinite blocklength, one can still aly ovasz ocal emma since the latter alies to a finite number of events A l B l, l. This is not the case when list size, and thus the number M of messages is infinite, since one has to emloy arguments different from this of comactness, to rove the existence of an -list ermutation invariant linear code. evertheless, the revious case deserves further investigation since according to the discussion in [9, Subsection II.B] and the references therein, transmission rates very close to the caacity H q of discrete memoryless symmetric channels can be treated, in case is very large. III. ERROR PROBABIITY AAYSIS Due to the channel symmetry, the average error list decoding robability Pe with constant list size of any linear code C, over the set of messages M equals Pe [, Aendix A]. Furthermore, for each code in the constructed ermutation ensemble E, message is encoded onto the all zero-codeword c so that Pry c, C = Pry c is valid for all C E. Taking the mean value over the ensemble E on both sides of 7 and utilizing the error decoding invariance rincile of emma we have min Pry c E [f de Ω y, C, λ, ]. P e λ, y Y Due to the concavity of the double exonential function f de over y Y where Ωy, C, λ,, alication of Jensen s inequality leads to min Pry c f de E [Ω y, C, λ, ]. P e λ, y Y Alying again Jensen s inequality to the convex function /x, of Ω y, C, λ, in 5 gives the following uer bound for Pe : min Pry c P e λ, y Fq f de Pry c λ m [Pry c E m, C λ]. 3 The comactness argument is utilized successfully in the roof of [3, Theorem 5..] The mean value aearing in the denominator of the double exonent in 3 is with resect to all ossible ermutations of the original code C. We extend here the bound of [4] to the case where an, R -list ermutation invariant linear code C defined on the field F q is transmitted over a discrete memoryless q-ary outut symmetric channel. emma 3: The mean value in the denominator of f de x in 3 is uer bounded for all as: E [ Pr y c m, C λ] M P l +...+l q =l q x F q vl,...,l q Q P Q Pr y x + 4 after setting λ = / + P and = P, where l,...,l q q, li =, i [, q ] v l,...,l q otherwise P, Q, S l,...,l q M P + =. 5 Q The roof of the lemma above is rovided in Aendix A. The numerator in is the multinomial coefficient. We next aly emma 3 to the right hand side of 3 in order to obtain the following analytic exression for the uer bound of an -list ermutation invariant code. The roof of the theorem is given in Aendix B. Theorem : Consider the transmission of an arbitrary set of messages M over a q-ary symmetric channel with the use of an, R -list ermutation invariant linear code C with M = q R, defined over the finite field F q. Then the average -list error decoding robability, over all messages in M, Pe of C is uer bounded as Pe min l l,p l P α,q l,, f de 6 M F C P β,q, where [ α,q l,, l q [ β,q, + + q q q and F C P l +...+l q =l l ] + vl,...,l q ] + 7 q P P P. 8

7 XEOUIS: IST PERMUTATIO IVARIAT IEAR CODES 7 Remark 6: Parameter P in Theorem is noted to control the effect of the distance between the code s weight distribution and this of a fully random block code. Each term of the weight distribution of the latter code equals the multinomial coefficient. For q = the bound of Theorem reduces to the bound [4, Theorem, eq.3]. Remark 7: The analysis rovided in the roofs of emma 3 and Theorem can be used in order to exand the generalization of the Shulman-Feder bound rovided in [, Aendix A] to the non-binary case. Remark 8: If the fixed list size is selected so that S l,l,...,l q, l, l,..., lq } v l,...,l arg max q. 9 l,...,l q then the otimum value of P in the bound 6 of Theorem, as well as in the analysis of [, Aendix A] which leads to the generalized form of Shulman-Feder bound, is P ot =. Then, we have F C lim F v l,...,l CP = max q. 3 P + l,...,l q In the sequel, we consider the limiting case P +, since it simlifies the analysis. Remark 9: Suose that for very large, there exists an, R binary linear code C that is -list ermutation invariant. Then, C cannot be fully random, i.e. the distance sectrum equals the binomial distribution. Otherwise we could aly inequality ex exx /x to the right hand side of 6 of Theorem for transmission over a binary symmetric channel and get an uer bound over the list error decoding robability that is tighter than the lower bound on the resective error robability rovided in [6]. A. Refinement of the bound of Theorem The effectiveness of the bound in 6 for an -list ermutation invariant linear code C is mainly affected by its weight distribution and the relation of the latter to the multinomial distribution. Indeed, the larger term F C in the denominator of the double exonent of 6 is, the looser the secific bound of Theorem becomes for the outut symmetric discrete memoryless channels. At least numerically, the roosed bound is noted to be loose for binary linear codes which contain the all codeword there exists a codeword in which exactly one symbol from F q \ } is reeated times. This also aears to be the reason Shulman-Feder bound is loose for some codes. A linear code C will not contain a codeword in which exactly one symbol from F q \ } is reeated times if and only if there does not exist a row in the arity check matrix of C which sums u to zero modulo q, where F q = q. In case C is binary, the revious statement is equivalent to the nonexistence of a row in the corresonding arity check matrix with an even number of ones. In what follows, we utilize Gallager s first bounding technique in order to rovide a tight bound for the resented class of codes, which enoy at least one codeword of Hamming weight. We consider an -list ermutation invariant linear code C that satisfies the conditions of emma for a non-negative integer t < / d min / and a ositive constant s such that s + t < / d min /. If we set C b c C : k F q \ }, c i = k i [, ]} 3 the error decoding region Y,C defined in 3 is equivalently exressed as the union of the following set Y,C b = y F q : l i } i=, with some c li C b such that Pry c li, C Pry c, C, l i } 3 and its comlement Ŷ,C b, where every c / C b. Consider a y Y,C b so that there exists a c C b with c i = k, i [, ] satisfying according to 4 If we define wty c wty. 33 t k y i : y i = k} 34 then for the considered y Y,C b and the secific codeword c where c i = k it holds according to 33 q q t l y t k y t l y t y t k y 35 where it is noted q t l y =, l= q t l y = wty. 36 l= In the secial case of F =, }, it holds and 35 gets the form t y = wty, t y = wty wty. Under the revious analysis and notations, the subset defined in 3 is equivalently exressed as Y,C b = y F q : l i } i= Pry c li, C Pry c, C, l i and t y t k y for some k F q \ }}. 37 Then the error robability P e P e Pry Ŷ,C b c + Pr is uer bounded as t y t k y} c 38 q k= where for y Ŷ,C b, t y > t k y, k F q \}. Due to the memoryless nature of the treated channels, the second term of the sum in the right hand side of 38 satisfies Pr = q q t y t k y} c = Pr t y t k y c k= q / k= l = q i= li= l l k l k= l,..., l q q Pri li 39 i=

8 8 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 where l i stands for the number of times symbol i aears in sequence y. Under the assumtion that the discrete memoryless channels are symmetric, the innermost sum in the right hand side of 39 equals l q l q i= li= l l k l. l,..., l k,..., l q Due to counting arguments it holds for every k Prt y t k y = Prt y t y. Thus 39 gets the form Pr q t y t k y} c = q k= or equivalently Pr q / l = q l q i= li= l l l t y t k y} c = q k= / l = l, where q l Û l, q = l l l,l q l l, q > l l,..., l q Û l l. 4 Regarding the first term of the sum in the right hand side of 38, the event y Ŷ,C b is equivalent to the statement that none of the codewords, that cause a list decoding error when y is the received sequence at the outut of the channel, belong to the set C b, defined in 3. Excluding set C b from the codewords that may cause a list decoding error does not alter the -list ermutation invariance roerty of the considered code C. Thus one can directly aly emma and the bounding technique that leads to Theorem to the first term of the sum in the right hand side of 38. Indeed, consider a vector y of Hamming weight w [w ot, / and let all codewords c such that wty c = wty = w i.e. we examine the articular column w of the code s distance distribution matrix Γ. Suose also there exists a c C b such that wty c = wty or equivalently there exists a k F q \ } where according to 34 and 36 it holds t y = t k y with t y = wty, q t k y = wty t l y or equivalently wty = + q t l y. l= l k l= l k Under our initial assumtion wty < /, the relation above and the set definition of Y,C b in 37 we deduce that there cannot exist a codeword c C b such that y Y,C b. Thus the exclusion of the set C b from the error decoding region Y does not alter the first / columns of the distance distribution matrix Γ. Consequently, C is still an - list ermutation invariant code. Indeed the first two conditions of emma 4 are satisfied, and this is also the case with the third one, since we have assumed s < / d min /. In accordance with Theorem the following lemma holds true. emma 4: Under the same assumtions of Theorem, the average -list error decoding robability, over all messages in M, Pe of an -list ermutation invariant linear code C, that satisfies conditions of emma for a non-negative integer t < / d min / and a ositive constant s such that s+t < / d min /, is uer bounded as P e min l l l f de M / l = Û l l α,q l,, ˆF C β +,q, l 4 q where functions α,q, β,q are defined in 7, Û l is defined in 4 and ˆF C v l,...,l max q. 4 l,...,l q l i<, i [,q ] Remark : The symmetric channel behavior encountered in the bounds of the current section is catured by the generic notions of symmetry in [] and []. In order to aly the double exonential technique to such symmetric channels, code structures different from the Hamming distance distribution matrix of Definition have to be considered, while artificial ensembles, ossibly different from the ermutation one, must be constructed. On the other hand the new bounds cannot be directly alied to non-symmetric channels. The lack of channel symmetry, as defined in the current work, revents from treating only the transmission of the all-zeros codeword c in the calculation of the error decoding robability. Furthermore, the ermutation invariance roerty as described in emma, is not necessarily reserved. Thus for the transmission of each message, we have to consider the largest error decoding region within the class of ermuted codes for the secific message, in order to aly the double exonential technique. The latter method is exected to lead to a loss of erformance in the deduced error bounds. IV. O THE REIABIITY FUCTIO OF BIARY MEMORYESS SYMMETRIC CHAES Through the alication of inequality f de x /x to the right hand side of 6, the list error decoding robability of an -list ermutation invariant linear code C satisfies for any

9 XEOUIS: IST PERMUTATIO IVARIAT IEAR CODES 9 where P e ex max Eo, q E o, q = ln q + ln R ln [ q + ln F } C q ] The domain of arameter in 43 motivates the quest for secific cases under which the revious bound aroaches from above and for very large block length the shere acking lower bound [6] on list decoding error robability for discrete memoryless symmetric channels. The latter bound is exressed as [ ex E o, q P e R ln O ]} + O 45 where is the otimum that maximizes the exonent in the right hand side of 45 and O = ln 8 + q ln 8, O = ln e + ln 8 min. Term min exresses the minimum transmission robability of a symbol. For the q-ary symmetric channel it holds min = /q. Under the scoe of 43 and 45 and the restriction that the fixed list size remains finite according to Remark 5, we are interested in investigating the existence of, R - list ermutation invariant linear codes for which F C in 3 is ln F C = o, where o/ aroaches as becomes large. In this way, we are able to rovide the reliability function of discrete memoryless symmetric channels for these rates under which Theorem is valid. In the current work, we restrict 3 to binary codes. Mimicking the arguments of [, Section II], we construct a random ensemble of, R binary linear codes by selecting indeendently, with the same robability each of the bits of the R lines of the generator matrix. In what follows, we denote by Dξ ξ the Kullback- eibler distance Dξ ξ ξ log ξ ξ + ξ log ξ ξ 46 while a. = b is used to denote the asymtotic exonential equivalence of terms a, b with resect to, as this is defined in [, Section II.A]. The weight distribution S ξ of an, R tyical binary linear block code from the 3 The main reason for not addressing non-binary codes in the current section is the fact that for very large blocklengths, one can only show code existence results with characteristics similar to these stated in 47. Thus a non-binary -list ermutation invariant linear code does not necessarily ossess a normal-like weight distribution for very large. linear code ensemble satisfies for very large and for every ɛ >.= R D ξ S, ξ ξ ξ R ɛ 47 =, ξ ξ R + ɛ where ξ R is the minimum distance of the code 4 rovided by the solution of Dξ R = 48 with resect to ξ. onbinary linear codes with weight distribution characteristics similar to these of 47 also aear in [3, Section 4]. Furthermore, the robability of an, R random linear code C over F with F being exonential large of order O is uer bounded as Pr F C e O Pr ξ : S ξ R D ξ +O. 49 The above relation is easily deduced with the aid of definitions in 5 and 3 resectively, if we emloy the following relations for very large Pr F C e O = and Pr Pr max <l ξ : S ξ S l R e O l R+O ξ Hξ o, D ξ = H ξ. ξ Combining 47 with 49, it is concluded that for very large the robability in the right hand side of 49 aroaches and thus a tyical linear code C from the random linear code. ensemble has ln F C = o. With the aid of 47 and Theorem, we examine the existence of codes from the linear code ensemble that are -list ermutation invariant. In articular, in the asymtotic blocklength regime, arameters S and S defined in and resectively satisfy in accordance with 47 for finite values of arameter s >, S. = ξ R R D + rc, S =. 5 Relacing 5 in Theorem, the sufficient condition 3 for the -list ermutation invariance roerty of a linear code satisfies equivalently min [ H ξ R ξ R H rc+d + rc ] R, [ ξ ]} R H < e. 5 4 In the asymtotic blocklength regime, linear codes C with rate R and relative minimum distance ξ R < / for which 48 is not valid do not exist.

10 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 It is noted that the necessary condition < S of Theorem, which enables the alication of ovasz ocal emma to the finite case, is satisfied since we consider finite list sizes, while S is exonentially large according to 5. Consequently, 5 is exressed with the aid of relation Dx / = H x as max D ξ R + D ξ R H r C R, H ξ R + r C } >. 5 Proosition : For all code rates R and relative covering radius r C under which 5 is valid, the reliability function of binary memoryless symmetric channels is rovided by shere-acking exonent ER su E o, R where E o, is given by 44. The calculation of the relative covering radius r C of a linear code C in the asymtotic blocklength regime is difficult and thus we resort to bounds that rovide sufficient conditions under which 5 is valid. As an examle, we consider the redundancy bound of [4, Proosition ] for a binary linear code C according to which r C R. The Kullback- eibler distance Dξ in 46 as well as Hξ are decreasing with resect to ξ. Thus, in case ξ R + R or R + ξ R 53 we have ξ R + r C and consequently 5 holds true for code rates R for which the following function 5 fr max D ξ R + D R ξ R } H R R, H ξ R 54 is strictly ositive. The secific function is illustrated in Fig. for those code rates R which satisfy 53. amely, for R.5475, fr is strictly ositive and monotonically increasing. Therefore, for all code rates R.5475 there exists a resective -list ermutation invariant binary linear code, whose error decoding robability coincides with shereacking lower bound. The revious rate region does not necessarily coincide with the critical region [5, eq ] for which the reliability function of the binary symmetric channel is known. For examle, let the binary symmetric channel with error transition robability =.5. Then, the reliability function of the channel equals su [,] E o, R for all rates R [R c 5 3, H 5 3 ], where R c is the critical rate of the channel given as R c log + + log + log In the binary case, Dx / and H x are symmetric around /. Thus D x / = Dx / and H x = H x. fr code rate R Fig.. Examination of the ositiveness of fr in 54 for code rates R which satisfy 53. Proosition extends the revious region to [.5475, H 5 3 ] since R c 5 3 = The roof of existence of -list ermutation invariant linear codes of finite blocklength with error decoding robability close to imroved shere acking bounds of [7] and [8] deserves further investigation as it requires an analytic construction rocess. In general, the roosed bound 43 is able to aroach the shere-acking bound 45 since the otimization arameter, that controls the magnitude of the sum of likelihood ratios /Ωy, C, λ, for a vector y, is not constrained to [, ]. To achieve this, secial structure is imosed on the examined codes as well as the artificially constructed ensembles, so that emma becomes valid and arameter ln F C in 8 is of order o. One can argue that the double exonential function substitutes in mitigating the effect of large values in E[/Ωy, C, λ, ] and thus can be greater than. V. COMPARISOS WITH PREVIOUS BOUDS: A EW CASS OF BOUDIG TECHIQUES The bounding technique which is resented in Section II and leads to bound 7 can be alied to general linear codes with no articular restrictions, such as list ermutation invariance. The resulting bound aears to be tighter than the DS bound for list decoding, but its comutability comes at the extra cost of requiring in-deth knowledge of the code s characteristics. Indeed, if we extend the sum in 7 over the set Fq, the average list error decoding robability of the linear code C over the set of messages M is uer bounded as P e min λ, y Fq f de Pry c Pry c λ. 56 m Pry c m λ

11 XEOUIS: IST PERMUTATIO IVARIAT IEAR CODES P e min λ, [,] g y:,} R g + g d r= d r g d=d min S d d g + g d r r λr d g fde. 63 Since the treated channels are memoryless and symmetric, we have in line to the roof of Theorem Pry c λ m Pry c m λ = q m λwty cm wty. 57 Furthermore, for a fixed y Fq, sequence wty c m } M m= equals the weight distribution of the coset that y belongs to, and is given by the iy-th row of matrix Γ according to Definition. Thus the right hand side of 57 equals λl wty Γ iy,l. 58 q If for each row i of the coset weight distribution matrix Γ we grou together those vectors y Fq of the same weight w and denote their number by Γ w i,l, the uer bound of 56 on Pe is exressed due to and the reasoning that leads to 88 in Aendix B as K w Pe min Γ w i,l λ, w q i= w= f de λl w Γ i,l. 59 q The DS bounding technique for discrete memoryless channels can be alied to the right hand side of 56 and thus lead to a bound similar to 59, which is tighter than the classical DS bound for list decoding. evertheless, its calculation requires details sequences Γ i,l and Γ w i,l about the structure of the linear code C, which are finer than these required for the exact calculation of error decoding robability. The above discussion identifies the difficulties in the exloration of new efficient bounding techniques, which rovide bounds tighter than the ones mentioned in the literature []. Identifying the order of summation in 56 with resect to y Fq and m [, M ] as the main reason that leads to the extremely comlicated bound 59, we are able to circumvent the revious difficulties and rovide a new tight bound. Thus, instead of following the analysis of 3-6, we note from 3 that for λ, if y Y,C, m λ Pry c f de. 6 Pry c m If we combine 6 with, we get the following bound on the error decoding robability for the secific linear code C Pe min Pry c λ, y Fq m λ Pry c f de Pry c m 6 where the sum with resect to y is exanded from Y,C to Fq. It is noted here that a direct comarison between 6 and the bound of 7 with the sum extended over Fq is not straightforward. Alying the DS bounding technique [, Sec. II.B] to the right hand side of 6 under the assumtion, we get for a non-negative un-normalized tilting measure G y Pe min λ G y Pry c [,] y Fq m y Fq Pry c G y fde Pry c Pry c m λ. 6 Due to inequality f de x /x, it follows that the right hand side of 6 is tighter than the classical DS bound and all its modifications related to Shulman-Feder bound, as reorted in [6] and []. In what follows we rovide an exression for the tight bound of 6 in the case of binary memoryless outut symmetric channels with F =, }, and G y = i= g y i for a ositive function g y :, } R. The general alhabet case is more comlicated and is rovided in Aendix C. The analysis for binary inut ternary outut has recently aeared in [7]. Theorem 3: Consider the transmission of a set of messages M with cardinality M with the use of an, R binary linear block code C over a binary symmetric channel with error transition robability. If the weight distribution of C is rovided by the sequence S d } d=d min where d min is the minimum distance of the code, then the average list decoding error robability over M is uer bounded as shown by 63 at the to of the current age. Proof: The base of the first term of the roduct in the right hand side of 6, which is raised to the ower, deends only on the transmitted codeword c and equals according to the assumtions of the theorem y F q G y Pry c = i= y i F q i= g y i Pry i =

12 IEEE TRASACTIOS O IFORMATIO THEORY, VO. XX, O. XX, XXXX 4 g y i Pry i = g y Pry y i F q y F q i= = g + g. 64 For the second term of the roduct in the right hand side of 6, we treat each term of the sum with resect to m searately. Secifically, consider a secific codeword c m of C with Hamming weight d and let the subsets I m and I m consist of the ositions in the sequence c m where symbols and aear resectively, so that I m = d and I m = d. Then, the double exonent in the right hand side of 6 satisfies λ Pry c = Pry c m i I m λ Pryi 65 Pry i and is indeendent of the ositions i I m. Thus, we have λ Pry c G y Pry c fde = Pry c m y Fq Pry i g y i i I m y i F q i I m Pry i g y i fde i I m Pryi Pry i i I m y i F q i I m λ. 66 The first roduct in the right hand side of 66 satisfies d = y F q Pry g y g + g d. 67 For the second term of the roduct in the right hand side of 66 consider a secific value of the subsequence y i } i I m which consists of r symbols and d r symbols. For this value, the resective summand in the examined term of the roduct amounts to d r r Pr g Pr g Pr λd r Pr f de Pr Pr λr d r g fde q = λr λd g r. 68 Since the right hand side of 68 is noted to deend only the Hamming weight r of the subsequence y i } i I m, d r subsequences contribute in the manner of 68 in the sum of the second term of the roduct in 66 and thus the latter sum equals d r= d r g d r r g λr λd f de.69 q The roof of the theorem is now comlete if in 6 we substitute 64 as well as the result of the combination of 69 with 67 along with the note that the roduct of these two terms is related to the codeword c m only through it s Hamming weight d. Remark : If we aly inequality f de x /x to the right hand side of 63, then one gets the DS bound for the binary symmetric channel. Indeed, if we set γ C g + g γ C, d g + g then we have P e γc d=d min S d γ C, d λd λd d d r= d r d r r λ g λ g = γ S d γ C, d λd λd d=d min d λ g + λ g = γ C S d γ C, d d=d min d λ λ g + λ λ g. Remark : Due to concavity of x, [, ], the bound of Theorem 3 can be alied to random ensembles of codes, where the average weight distribution is easier to calculate than in the stand-alone case. Remark 3: Comaring the double exonential function f de x with f x /x for x >, as illustrated in Fig., f de x is noted to mitigate the effect of codewords c m in the sum 6 for which the likelihood ratio Pry c / Pry c m < wty c wty < resectively for a given y Fq. The question whether a horizontally fli sigmoid function different from the double exonential function and strictly lower than or equal to f x can be emloyed in the DS bounding technique naturally arises. Sigmoid functions interreted as cumulative distribution functions of continuous robability functions aear to be good candidates. In the sequel we consider the normalized horizontally fli sigmoid f x erfc.5e x 7 erfc.5 deicted in Fig., where erfcx = π x e z dz. The scaling factor.5 is selected so that f x is as close as ossible to f x in the region close to. We refer to the uer bound 63 where f de x is relaced by f x as 63-erfc. It is noted that f x is slightly larger than f x in the region